ELLIPTIC CURVES AND STABLE HOMOTOPY I, PRE-RELEASE VERSION: February 22, 1999

M. J. HOPKINS AND H. R. MILLER

Abstract. In this note a new cohomology theory will be constructed. From the point of view of homotopy theory, it is the most natural candidate for elliptic cohomology. The construction is via obstruction theory.

Contents

2 M. J. HOPKINS AND H. R. MILLER

1. Introduction 3 2. A_{∞}-structures and maps 7 2.1. A_{∞}-ring spectra 7 2.2. The moduli spaces in a useful case 7 3. Lifting diagrams 11 3.1. The realization space 11 3.2. Centric diagrams 11 4. Quillen (co-)homology 13 4.1. Generalities 13 5. Quillen associative algebra (co-)homology 15 5.1. Associative algebras 15 5.2. Commutative rings 17 5.3. Schemes 17 5.4. Vanishing criteria 19 6. Stacks 21 6.1. The Čech complex 23 6.2. Inverse limits and Čech cohomology 23 6.3. The quasi-etale topology 25 6.4. The associated stack 25 6.5. The cotangent complex of a stack 25 7. The moduli stack of formal groups 27 7.1. The Frobenius morphism 27 8. Elliptic curves 29 8.1. The moduli stack 29 8.2. Height and the Frobenius isogeny 31 8.3. The invariant differential 31 8.4. Representability of M_{Ell} → M_{FG} 33 8.5. Some vanishing results 35 8.6. A quasi-etale cover 35 9. Determination of certain cotangent complexes 41 9.1. The elliptic moduli stack 41 9.2. The cotangent complex of M_{Ell} over M_{FG} 41 10. Elliptic spectra 43 11. Factoring and lifting 45 11.1. The section (M_{Ell})′_{et} → ho A_{∞} 45 11.2. The lift to A_{∞} 45 12. Computations of Quillen cohomology 49 12.1. Computations modulo a prime power 49 12.2. Computations over a ℚ-algebra 53 12.3. Computations over ℤ 55 13. Coherent cohomology of the stack M_{Ell} 57 13.1. When 6 is invertible 57 13.2. When 3 is nilpotent, and all geometric fibers are

ordinary 59

13.3. When 2 is nilpotent and all geometric fibers are

ordinary 59

Appendix A. Low dimensional cohomology of categories

63

Appendix B. A spectral sequence for A_{∞}-maps 69 B.1. The spectral sequence 69 B.2. The Reedy model category structure 71 B.3. The P model category structure 71 B.4. The F^{α} model category structure 73 Appendix C. A calculation: so I don’t have to keep

reconstructing this 77

Appendix D. Temporary ramblings to work something

out 79

Appendix E. Calculation 79 Appendix F. Equivalence of realization spaces 81 References 85

ELLIPTIC COHOMOLOGY (February 22, 1999) 3

1. Introduction

The point of this document is to make use of A_{∞} moduli spaces to rigidify the diagram of etale elliptic spectra. The main theorem is

Theorem 1.1. Let M_{Ell} be the moduli stack of generalized elliptic curves, and (M_{Ell})_{et} the category of etale open subsets of M_{Ell}. There is a presheaf

E : (M_{Ell})_{et} → A_{∞}^{Ell}

with values in the category of cofibrant-fibrant A_{∞} elliptic spectra (see below) which has cohomological descent and, when restricted to (M_{Ell})′_{et} comes equipped with an isomorphism of functors

spec π_{0}E(U) ≈ 1.

Any two such presheaves are naturally weakly equivalent.

Definition 1.2. An elliptic spectrum is a triple (E, A, t) in which

(1) E is a multiplicative spectrum with the property that π_{∗}E commutative, concentrated in even degrees, and π_{2}E contains a unit; (It follows that E is complex orientable and has a formal group G canonically defined over π_{0}E.)

(2) A is a generalized elliptic curve [3] over π_{0}E

(3) t : G → A^{f} is an isomorphism of the formal group G with the formal completion of A.

The elliptic spectra form a category in which a map

f : (E, A, t) → (E′, A′, t′)

consists of a multiplicative map f_{1} : E → E′, and a map of generalized elliptic curves f_{2} : A′ → π_{0}^{∗}A such that the following diagram commutes

G′ ⟶ π_{0}f^{∗}G
t↓  ↓_{t′}
(E′)^{f} ⟶_{(f_{2})^{f}} π_{0}f^{∗}E.

Let M_{Ell} be the moduli stack of generalized elliptic curves with integral geometric fibers (§ 6). An elliptic spectrum (E, A, t) comes equipped with a map

spec π_{0}E → M_{Ell}

classifying the elliptic curve A.

Definition 1.3. An elliptic spectrum (E, A, t) is etale if the map spec π_{0}E → M_{Ell} is open subset of M_{Ell} is the etale topology.

For a stack M let (M)_{et} be the category of etale open subsets of M of finite type. Attaching to each etale elliptic spectrum the map classifying its elliptic curve thus defines a functor from the category of etale elliptic spectra to the category (M_{Ell})_{et}. One goal of this paper is to “reverse” this process and construct a kind of sheaf on (M_{Ell})_{et} with values in A_{∞} ring spectra.

Let A_{∞}^{Ell} be the (topological) category consisting of triples (E, A, t) consisting of an A_{∞} ring spectrum E together with an etale elliptic structure on its underlying spectrum, and whose morphism space from (E_{1}, A_{1}, t_{1}) to (E_{2}, A_{2}, t_{2}) is the fiber product

A_{∞}^{Ell} ((E_{1}, A_{1}, t_{1}), (E_{2}, A_{2}, t_{2})) ⟶ S^{Ell} ((E_{1}, A_{1}, t_{1}), (E_{2}, A_{2}, t_{2}))
↓  ↓
A_{∞} (E_{1}, E_{2})  ⟶  S (E_{1}, E_{2}) .

Then the process spec π_{0} defines a functor

ho A_{∞}^{Ell} → (M_{Ell})_{et} .

Occasionally the symbol spec π_{0} will also refer to the composite functor A_{∞}^{Ell} → (M_{Ell})_{et}.

Let (M_{Ell})′_{et} be the subcategory of (M_{Ell})_{et} consisting of affine U → M_{Ell} for which the sheaf ω_{U} admits a nowhere vanishing global section (ie the restriction of the line bundle ω to U is trivializable). Our first main result is that the functor spec π_{0} admits a more or less canonical section over (M_{Ell})′_{et}

(M_{Ell})′_{et} → ho A_{∞}^{Ell}.

ELLIPTIC COHOMOLOGY (February 22, 1999) 5

Next we will apply the Dwyer-Kan lifting machinery to the the problem

A_{∞}
↓
(M_{Ell})′_{et} ⟶ ho A_{∞}^{Ell} ⟶ ho A_{∞}.

We will see that the space of lifts is non-empty and connected, thus giving an essentially unique diagram. This diagram of spectra is elliptic cohomology.

Working remark 1. Work in the ommission of additive fibres.

2. A_{∞}-structures and maps

2.1. A_{∞}-ring spectra. We will use the theory of spectra as developed by May et. al. Let A_{∞} denote the category of A_{∞}-ring spectra indexed on a fixed universe U. This is a closed topological model category and is generated by small objects [?]. The functor

X ↦ T X = ⋁ L(n) ⋊ X^{(n)},

is a monad (or triple), and an A_{∞}-ring spectrum is an algebra over this monad.

Let A_{∞} denote the category of A_{∞} ring spectra. There are adjoint functors

T : S ⇄ A_{∞} : F,

where by a mild abuse of notation, the functor T is regarded as taking it’s values in A_{∞}, and F is simply the forgetful functor. As the diagram suggests, the functor T is left adjoint to F. The morphism objects of A_{∞}, being subsets of the morphism objects of S are naturally topological spaces.

The following result is proved in [?]. It also holds with A_{∞} replaced by M-algebras, where M is any operad over the linear isometries operad.

Theorem 2.1. i) The category A_{∞} is canonically enriched over topological spaces. In other words, there are bifunctors

(S, X) ↦ S ⊗ X : 𝔗 × A_{∞} → A_{∞}
(Y, S) ↦ Y^{S} : A_{∞} × 𝔗^{op} → A_{∞}

and natural homeomorphisms

A_{∞}(X, Y^{S}) ≈ A_{∞}(S ⊗ X, Y) ≈ 𝔗(S, A_{∞}(X, Y)).

ii) The category A_{∞} is cartesian closed in the sense of enriched categories: it contains all small indexed limits and colimits;

iii) The category A_{∞} has a unique closed model category structure in which a map X → Y is a weak equivalence (resp. fibration) if and only if the map F X → F Y is. This model category structure is generated by small objects and is a topological model category in the sense that Quillen’s axiom SM7 [6] holds, with “simplicial set” replaced by “toplogical space.”

ELLIPTIC COHOMOLOGY (February 22, 1999) 7

iv) If X_{•} is a simplicial object in A_{∞}, then the natural map

|F X_{•}|_{S} → F |X_{•}|_{A_{∞}}

is an isomorphism.

(An indexed (co-)limit is a (co-)limit over a topological category. The basic example is the geometric realization of a simplicial space.)

Definition 2.2. Suppose that X is a spectrum. The space A_{∞} {X} is the nerve of the category whose objects are weak equivalences F (R) → X, and whose morphisms are commutative squares of weak equivalences

F (R) ⟶^{F (f)} F (S)
↓  ↓
X  X.

In other words, if (A_{∞})_{w} denotes the topological category of A_{∞} ring spectra and weak equivalences, then A_{∞} {X} is the nerve of the category of (A_{∞})_{w} objects over X.

In case X comes equipped with a homotopy associative multiplication μ we will let A_{∞} {X}_{μ} denote the nerve of the full subcategory with objects f which are maps of homotopy associative ring spectra. The space A_{∞} {X} is the disjoint union of the spaces A_{∞} {X}_{μ} as μ ranges over the different homotopy associative multiplications on X (monoid structures on X in ho S), since the same is true for the over-category. In case the multiplication μ is an integral part of the ambient structure of X (like when X is elliptic) the subscript will be dropped from the notation.

Rezk has shown that the space A_{∞} {X} is weakly equivalent to the space of operad maps from the A_{∞} operad to the endomorphism operad of X.

Working remark 2. Define ring spectrum to mean spectrum equipped with a homotopy associative multiplication.

2.2. The moduli spaces in a useful case. We are interested in determining the homotopy types of the spaces A_{∞} {X}_{μ} and A_{∞}(E, F). It turns out that a lot can be said about these spaces in the presence of certain special conditions. Suppose that F is a ring spectrum satisfying the following condition of Adams [1, Condition 13.3 (page 284)]:

Condition 2.3. Each map of spectra

X → F

with X finite, factors as

X → V → F

where V is finite and both F_{∗}V ane F^{∗}V are projective over F_{∗}.

Examples of spectra satisfying this abound. In fact

Proposition 2.4. Condition 2.3 holds if F is Landweber exact.

Proof: It is easy to check the condition for MU (see [1, Prop. 13.4]), and the proof applies to any spectrum having the homotopy type of a wedge of suspensions of MU. If F is Landweber exact, then the map

V = ⋁_{x∈π_{∗}F} Σ^{|x|}MU → F

gives, for any Z a surjective map

V_{∗}Z → F_{∗}Z.

It follows using Spanier-Whitehead duality that if X is finite, any map X → F factors through V.

Working remark 3. Clean up the above proof. Work out the spectral sequences in the moduli space case. Also, you better either really use the Adams condition, or be explicit about how you are modifying it.

Proposition 2.5. Suppose that (F, μ) is a ring spectrum satisfying Condition 2.3. There is a spectral sequence

E_{2}^{s,t} ⟹ π_{t−s}(A_{∞} {F }_{μ} , f),

with

E_{2}^{s,t} = \begin{cases}Der_{π_{∗}F}^{s+1} (F_{∗}F, F̃^{0}S^{t}) & s ≥ 0, t ≥ 1 \\ ∗ & (s, t) = (0, 0) \\ undefined & otherwise \end{cases}

ELLIPTIC COHOMOLOGY (February 22, 1999) 9

The group Der^{s}( − , −) is Quillen’s sth derived functor of associative algebra derivations, and is described in § 5.

Proposition 2.6. Suppose that E and F are A_{∞} ring spectra, F satisfies Condition 2.3, and f : E → F is a homotopy class of maps of the underlying ring spectra. There is a spectral sequence

E_{2}^{s,t} ⟹ π_{t−s}(A_{∞}(E, F); f̃)

with

E_{2}^{s,t} = \begin{cases}Der_{F_{∗}}^{s} (F_{∗}E, F̃^{0}S^{t}) & t ≥ 1 \\ ∗ & (s, t) = (0, 0) \\ undefined & t = 0, s > 0 \end{cases}

3. Lifting diagrams

This section consists of a collection of material from [4].

3.1. The realization space. Let S be a closed simplicial (or topological) model category, and I a small category. The situation relevant to this paper is that I is the opposite category of (M_{Ell})′_{et} (see § ??), and S is the category A_{∞}. We suppose given an Idiagram in ho S (ie, an object of (ho S)^{I}, and we want to consider the problem of lifting it to an object of S^{I}. This problem can be attacked by studying the realization space r(X) of X.

Definition 3.1. With the above notation, the realization category of X is category whose objects are pairs (Y, f) consisting of an object Y of S^{I}, together with an isomorphism f : πY → X in (ho S)^{I}. A morphism in r(X) from (Y_{1}, f_{1}) to (Y_{2}, f_{2}) is a map g : Y_{1} → Y_{2} (necessarily a weak equivalence) making the following diagram commute

πY_{1} ⟶^{πg} πY_{2}
f_{1}↓  ↓^{f_{2}}
X  X.

The realization space, is the classifying space of the realization category. Both the category and its classifying space are denoted r(X).

There is another useful space to consider in this regard. Let lift(X) be the nerve of the category whose objects are diagrams Y with the property that πY is isomorphic to X, and whose morphism are weak equivalences. There is a fibration

r(X) → lift(X) → B Aut(X).

By making a simplicial resolution of I by free categories, the spaces r(X) and lift(X) can be expressed as the total complexes of cosimplicial spaces which we will call r(X)_{•} and lift(X)_{•} respectively. While the cosimplicial spaces depend on the choice of resolution, the homotopy spectral sequence is independent of the resolution from the E_{2}-term on. There is a termwise fibration

r(X)_{•} → lift(X)_{•} → B Aut X

ELLIPTIC COHOMOLOGY (February 22, 1999) 11

where B Aut X is thought of as a constant cosimplicial space. The homotopy spectral sequences for r(X)_{•} an lift(X)_{•} therefore coincide except possibly in the E_{2}^{0,0}-term where the latter is the quotient of the former by the action of π_{0} haut(X). In fact, the spectral sequences coincide anyway, since before the spectral sequence can be defined, one must choose a component of the 0-space which is equalized by d^{0} and d^{1}, and then restrict to this component.

Working remark 4. Write an appendix about the homotopy spectral sequence of a non-pointed cosimplicial space.

3.2. Centric diagrams. A diagram X as above is said to be centric if for each morphism f : d → d′ in I, the map

S(Xd′, Xd′)_{1} → S(Xd, Xd′)_{f}

is a weak equivalence ([4]). In the case of a centric diagram there is a simple description of the E_{2}-term of the spectral sequence.

Proposition 3.2. [4, §3] Let X : I → ho S be a centric diagram. There is a spectral sequence

E_{2}^{s,t} ⟹ π_{t−s} r(X) ,

with

E_{2}^{s,t} = \begin{cases}\underleftarrow{lim}^{s}π_{t} Bhaut_{1} X( −) & s ≥ 0 and t ≥ 2 \\ \{1\} & (s, t) = (0, 1) \\ ∗ & (s, t) = (0, 0) or (1, 1) \\ undefined & otherwise. \end{cases}

Working remark 5. I talked to Bill. I need to indicate that it is only the identity component of haut(X) that comes in, so that the spaces Bhaut_{1}(X) are, in fact, simply connected.

4. Quillen (co-)homology

4.1. Generalities. Fix a ground ring k and let i : A → B be a map of k-algebras. Quillen defines a homology object L_{B/A} as follows. First he shows that the category of simplicial A-algebras is a closed model category when one defines weak equivalence to be quasi-isomorphism, and one takes all objects to be fibrant. To define the homology object, regard B and A as constant simplicial objects, and let A → P_{•} → B be a factorization into a cofibration followed by a weak equivalence. The complex L_{B/A} is then defined by

L_{B} = Ω_{P_{•}/A}^{1} ⊗_{P_{•}} B.

The quasi-isomorphism class of L_{B/A} is independent of the resolution.

The association B/A ↦ L_{B/A} is functorial: Given a map of pairs

A ⟶^{f} B
i↓  ↓^{j}
C ⟶_{g} D

there is a natural map L_{j/i} : j^{∗}L_{B/A} → L_{D/C}. The map associated to a composite of maps of pairs is chain homotopic to the composite of the maps associated to each pair.

Actually, for fixed A, the association L_{B/A} can be made into a functor with values in the category of chain complexes of Bmodules. To do this one makes use of the fact the the category of simplicial A-modules is generated by small objects, and so the factorization into a cofibration followed by acyclic fibration can be made functorially. We will make use of this rigidified functoriality. One can also simply use the standard resolution based on the triple “free A-algebra generated by the underlying point set of B.”

We will often write the absolute homology L_{B/k} as simply L_{B}.

Given a sequence A → B ⟶^{f} C Quillen shows that the natural sequence

f^{∗}L_{B/A} → L_{C/A} → L_{C/B}

ELLIPTIC COHOMOLOGY (February 22, 1999) 13

is a cofibration. This cofibration is known as the transitivity sequence. It is useful to write down this sequence in the case A = k (which is no loss of generality, since no use has been made so far of k):

f^{∗}L_{B} → L_{C} → L_{C/B}.

It shows that the relative homology object is the cofiber of the map of absolute homology objects, and that the homology of the transitivity sequence can be thought of as the long exact sequence of a pair.

Another important result is the following..

Proposition 4.1 (Flat base change). Suppose that

A ⟶^{f} B
i↓  ↓^{j}
C ⟶_{g} D

is a pushout square with the property that the groups Tor_{t}^{A}(C, B) are 0 for t > 0. Then the natural map g^{∗}L_{B/A} → L_{D/C} is an equivalence.

Note that the condition on Tor holds if either i or f is flat.

Proof: Factor f as A → P_{•} → B into a cofibration followed by an acyclic fibration. Then C → i^{∗}P_{•} = C ⊗_{A} P_{•} is a cofibration, and the natural map i^{∗}P_{•} → D is a weak equivalence since the s^{th} homology group of i^{∗}P_{•} is Tor_{s}^{A}(C, B). The result follows easily from this.

The transitivity sequence and the flat base change result combine to give a Meyer-Vietoris sequence.

Corollary 4.2. Suppose given a pushout square as in Proposition 4.1. Then the square

(jf = gi)^{∗}L_{A} ⟶^{j^{∗}L_{f}} j^{∗}L_{B}
g^{∗}L_{i}↓  ↓^{L_{j}}
g^{∗}L_{C}  ⟶_{L_{g}} L_{D}

is a homotopy pushout. It follows that there is a cofibration

L_{A} ⟶^{[j^{∗}L_{f} ,g^{∗}L_{i}]^{T}} j^{∗}L_{B} ⊕ g^{∗}L_{C} ⟶^{[−L_{j},L_{g}]} L_{D}.

Proof: The map of transitivity cofibrations for f and g can be written as

j^{∗}f^{∗}L_{A} ⟶ j^{∗}L_{B} ⟶ j^{∗}L_{B/A}
↓  ↓  ↓^{L_{j/i}}
g^{∗}L_{C} ⟶ L_{D} ⟶ L_{D/C}.

Proposition 4.1 implies that the map L_{j/i} is an equivalence. The result follows.

5. Quillen associative algebra (co-)homology

The basic reference for this section is [6].

5.1. Associative algebras. For each map of associative algebras A → B, let

D_{B/A}

be the complex of B bi-modules which is the Quillen homology object of B/A. In case B is flat over A (and projective as a module), this complex is weakly equivalent to the complex

\begin{aligned}B & ⊗_{A} B^{op} \\ & ↓ \\ & B \end{aligned}  \begin{matrix}0 \\ −1 \end{matrix}

The object D_{B/A} is functorial in B/A in the sense that a diagram

\begin{matrix}A & ⟶ & A′ \\ ↓ & & ↓ \\ B & ⟶_{f} & B′ \end{matrix} (5.1)

gives rise to a map (f ⊗ f)^{∗}D_{B/A} → D_{B′/A′}. Quillen shows that given,

A → B ⟶^{f} C

ELLIPTIC COHOMOLOGY (February 22, 1999) 15

the sequence

f^{∗}D_{B/A} → D_{C/A} → D_{C/B}

is a cofibration. The long exact sequences derived from this cofibration are known as transitivity sequences.

Working remark 6. Actually, Quillen doesn’t mention this sequence, but it is easy to derive. It is a little misleading, though. See the discussion of the case of commutative rings below.

Another important tool is the following base change result.

Lemma 5.2. Suppose A is a commutative and that B is a flat associative A-algebra containing A in its center. For any map A → A′, write

B′ = B ⊗_{A} A′,

and let j be the map

j : B ⊗_{A} B → B′ ⊗_{A′} B′
b_{1} ⊗ b_{2} ↦ b_{1} ⊗ 1 ⊗ b_{2} ⊗ 1.

Then the map (5.1)

j^{∗}D_{B/A} → D_{B′/A′}

is a weak equivalence.

Proof: To ease the notation, the symbol ⊗, unless indicated otherwise, will refer to ⊗_{A}. Since B/A is flat, so is B′/A′, and we can make use of the cofiber sequences

D_{B/A} → B ⊗ B → B

D_{B′/A′} → B′ ⊗_{A′} B′ → B′.

Let P_{·} → B be a projective resolution of B as a B ⊗ B-module. Then

P ⊗_{B⊗B} (B′ ⊗_{A′} B′) ≈ P ⊗_{B⊗B} (B ⊗ B ⊗ A′) ≈ P ⊗ A′ (5.3)

is a complex of projective B′ ⊗_{A′} B′-modules, and it’s homology is

Tor_{A}^{∗}(B, A′) = \begin{cases}B′ & if ∗ = 0 \\ 0 & otherwise. \end{cases}

It follows that (5.3) is a projective resolution of B′. Lifting the map B ⊗ B → B to P_{·} gives a cofibration

D_{B/A} → B ⊗ B → P_{·},

and hence a map of cofibrations

j^{∗}D_{B/A} ⟶ B′ ⊗_{A′} B′ ⟶ j^{∗}P_{·}
↓  ↓  ↓
D_{B′/A′} ⟶ B′ ⊗_{A′} B′ ⟶ B′.

Since the second and third vertical maps are weak equivalences, so is the first. The result follows.

5.2. Commutative rings. From now on suppose that A → B is a map of commutative rings. In this case we define a complex of B-modules

Λ_{B/A} = μ^{∗}D_{B/A} = B ⊗_{B⊗B}^{L} D_{B/A}.

The L indicates that the tensor product is taken in the homotopy invariant sense, so that, even though the homology of the complex D_{B/A} is concentrated in dimension zero, the homology of Λ_{B/A} need not be. An sequence A → B → C gives rise to a cofibration in the homotopy theory of chain complexes of B-modules.

f^{∗}Λ_{B/A} → Λ_{C/A} → Λ_{C/B}.

Working remark 7. This is the misleading expression. It is important to remember that μ is the map from C ⊗_{A} C to C (as opposed to the map from C ⊗_{B} C to C.) The consequence of this is that, in the case where both maps are flat, the homology groups of μ^{∗}Λ_{C/B} are not the groups Tor_{i−1}^{C⊗_{B}C}(C, C). Of course, as in the comment below, if the complex D_{C/B} is acyclic (as in the application below), then the complex Λ_{C/B} is also acyclic.

Note that if C = S^{−1}B, for some multiplicatively closed subset S ⊂ A, then D_{B/C} is acyclic, and so by the above exact sequence, the maps

\begin{aligned}(S × S)^{−1}D_{B/A} & → D_{C/A} \\ S^{−1}Λ_{B/A} & → Λ_{C/A} \end{aligned} (5.4)

ELLIPTIC COHOMOLOGY (February 22, 1999) 17

are isomorphisms.

Working remark 8. Is the above what I need, or is it enough to deduce what I need from the flat base change result earlier.

Lemma 5.5. Suppose that P is a complex of R-modules and s ∈ R an element which is not a divisor of zero. Let π : R → R/(s) be the quotient map. If π^{∗}P is contractible, then

P → s^{−1}P

is a weak equivalence.

Proof: The complex s^{−1}P is the colimit of

P ⟶^{s} P ⟶^{s} · · · .

Since s is not a divisor of zero, the cofiber of P ⟶^{s} P is π^{∗}P. Since π^{∗}P is contractible, each map in the above sequence is a weak equivalence. It follows that the map P → s^{−1}P is a weak equivalence since homology commutes with directed colimits.

Remark 1. This can be considerably generalized, but this version is so simple, and is all that is needed in the present paper.

5.3. Schemes. Suppose now that X/S is a flat scheme, and define complexes

D_{X/S} = (O_{X} ⊗_{O_{S}} O_{X} → O_{X}) [−1]
Λ_{X/S} = RΔ^{∗}D_{X/S}.

The complex D_{X/S} is a complex of coherent sheaves on X × X, and Λ_{X/S} is its (derived) pullback to the diagonal. By the isomorphism (5.4) if i : U ⊂ X is the inclusion of an open subscheme, the restriction maps

i^{∗}D_{X/S} → D_{U/S}
i^{∗}Λ_{X/S} → Λ_{U/S}

are weak equivalences.

In the context of schemes, the fundamental exact sequences take the following form. Given

X ⟶^{f} Y ; ↓ ; S

there are cofibrations

\begin{matrix}(f × f)^{∗}D_{Y/S} → D_{X/S} → D_{X/Y} \\ f^{∗}Λ_{Y/S} → Λ_{X/S} → Λ_{X/Y} . \end{matrix} (5.6)

As far as computation goes, we have the following result.

Lemma 5.7. If X/S is smooth, there is an isomorphism

H_{i}Λ_{X/S} ≈ Ω_{X/S}^{i+i}

Proof: The question is local on X/S (provided we make a map) and so we reduced to the case that X = A_{S}^{n}. In this case one can proceed by direct calculation. I’ll fix this later. It also follows from Quillen’s spectral sequence, which degenerates in this case since there is no room for differentials, and no room for extensions (I think).

Lemma 5.8. Suppose that X is a scheme, j : Z ⊂ X is a closed subscheme, and i : U ⊂ X its complement. If P is s complex of coherent O_{X}-modules and j^{∗}P is acyclic, then the map

P → i_{∗}i^{∗}P

is a weak equivalence.

Lemma 5.9. Suppose that X is a scheme j : Z ⊂ X is a closed subscheme, and P a complex of coherent O_{X}-modules. If each of the sheaves H_{i}P is supported on Z and j^{∗}P is contractible, then P is contractible.

Proof: ???I don’t need this, but I think it is true.???

ELLIPTIC COHOMOLOGY (February 22, 1999) 19

5.4. Vanishing criteria.

Proposition 5.10. Let A → B be a map of commutative rings. The complex of B-modules Λ_{B/A} is acyclic if and only if the complex L_{B/A} is acyclic.

Proof: The ss.

Now suppose that A and B are algebras over ℤ/p. Let ϕ be the absolute Frobenius map

\begin{matrix}B & ⟶^{ϕ} & B \\ ↑ & & ↑ \\ A & ⟶^{ϕ} & A \end{matrix} ϕ(x) = x^{p}

and

ϕ_{B/A} : ϕ^{∗}B → B

the relative Frobenius. More generally, write ϕ_{B/A}^{n} : (ϕ^{n})^{∗}B → B for the iterated relative Frobenius.

Lemma 5.11. For each n > 0, the map

L_{((ϕ^{n})^{∗}B)/A} → L_{B/A},

induced by the iterated relative Frobenius, is null.

Proof: The result is clear if B is a polynomial algebra over A, since Dx^{p^{n}} = 0. It follows that if P_{•} is a simplicial A-algebra, which is dimension-wise a polynomial algebra, then the map

L_{ϕ^{∗}P_{•}/A} → L_{P_{•}/A},

is zero. Taking P_{•} → B to be a resolution gives the desired result.

Corollary 5.12. If for some n, the map ϕ_{B/A}^{n} is an isomorphism, then L_{B/A} and D_{B/A} are acyclic.

Proof: With these assumptions, the relative Frobenius map

L_{ϕ^{∗}B/A} → L_{B/A},

is both an isomorphism and null. It follows that L_{B/A} is acyclic. The fact that D_{B/A} is acyclic follows from Proposition 5.10.

ELLIPTIC COHOMOLOGY (February 22, 1999) 21

6. Stacks

Let C be a category with a Grothendieck topology.

Definition 6.1. A presheaf of groupoids on C is a rule associating to each object X of C a groupoid 𝔊_{X}, to each map f : X → Y a functor 𝔊f = f^{∗} : 𝔊_{Y} → 𝔊_{X} and to each composable pair

X ⟶^{f} Y ⟶^{g} Z

a natural transformation ϵ(f, g) : f^{∗}g^{∗} → (g ∘ f)^{∗}, satisfying the associativity condition

ϵ(ϵ(f, g), h) = ϵ(f, ϵ(g, h)). (6.2)

The collection of presheafs of groupoids over C forms a 2-category in which the morphisms

Now suppose 𝔊 is a presheaf of groupoids on C and that U → X is a covering. The category Desc(U/X) of descent data relative to U is the category whose objects are objects E of 𝔊(U) together with an isomorphism s : π_{1}^{∗}E → π_{2}^{∗}E satisfying the cocycle condition

fill this in.

A presheaf of groupoids 𝔊 is a sheaf if for each covering U → X of C the map

𝔊(X) → Desc(U/X)

is an equivalence of groupoids.

Definition 6.3. A stack in C is a presheaf of groupoids.

Given a stack 𝔊 over C one can form a category 𝔈𝔊 in which an object consists of a pair (E, X) with X an object of C and E an object of 𝔊(X). A map from (E_{1}, X_{1}) to (E_{2}, X_{2}) consists of a map f : X_{1} → X_{2} together with a morphism g : E_{1} → f^{∗}E_{2}. It is convenient to picture this data as

E_{1} ⟶^{g} E_{2}
↓  ↓
X_{1} ⟶_{f} X_{2}.

The fact that 𝔈𝔊 is a category makes use of the associativity condition 6.2. The category 𝔈𝔊 is fibered in groupoids over C.

The collection of stacks over C forms a 2-category, in which a map is ...

A 1-morphism M_{1} → M_{2} of stacks is said to be representable if for each 1-morphism X → M_{2} with X representable, the fibre product

X ×_{M_{2}} M_{1}

is representable.

If P is a property of morphisms in C which is stable under change of base, and which is local in nature, then a 1-morphism M_{1} → M_{2} will be said to have property P if it is representable, and if for each X → M_{2}, the map

X ×_{M_{2}} M_{1} → X (6.4)

has property P. One important example is the case in which P is the property of being a covering.

We can define a kind of Grothendieck topology on the 2-category of stacks by letting the covers be the 1-morphisms which are coverings in the above sense.

Working remark 9. It might be more correct to work with the “sieve” formulation since, strictly speaking, we don’t wish to work with fibre products, but only “homotopy fibre products.”

By construction, the coverings of StC satisfy the following axioms:

(1) If T_{1} → T_{2} is a representable equivalence then it is a covering;

(2) If T_{1} → T_{2} is a covering, and U → T_{2} is any 1-morphism, then the 1-morphism U ×_{T_{2}} T_{1} → U is a covering;

(3) The class of coverings is stable under formation of coproducts and composition.

A presheaf on StC is a contravariant functor from StC to 𝔖𝔢𝔱𝔰. A presheaf on a stack M is a presheaf on the 2-categry of stacks in C over M. Here we must regard 𝔖𝔢𝔱𝔰 as a 2-cagegory in which the only 2-morphisms are identity maps. This has the consequence that if P is a presheaf, and f and g are two 1-morphisms which are related by a 2-morphism, then P f = P g.

ELLIPTIC COHOMOLOGY (February 22, 1999) 23

A presheaf is a sheaf if it satisfies the sheaf axiom: For each covering T_{1} → T_{2}, the sequence

P (T_{2}) → P (T_{1}) ⇉ (T_{1} ×_{T_{2}} T_{1})

is an equalizer.

6.1. The Čech complex. Suppose that p : M_{1} → M_{2} is a morphism of stacks, and that F is a sheaf on M_{2}.

1. objects of C

2. maps X → M

3. equivalence?

4. cohomology?

6.2. Inverse limits and Čech cohomology. Again, let C be a category with a Grothendieck topology, and F a presheaf of abelain groups. The point of this section is to identify the groups

\underleftarrow{lim} _{C′}^{s}F

where C′ ⊂ C is a full subcategory. We are interested in full subcategories generated by collections U ⊂ ob C which are coverings of the terminal object in the sense that for each X ∈ ob C, the map

∐_{U∈U} U × X → X

is a covering. The Čech complex of U is defined to be the normalized complex associated to the cosimplicial abelian group

∏_{U∈U} F (U) ⇉ ∏_{(U_{0},U_{1})∈U^{2}} F (U_{0} × U_{1}) ….

The cohomology of the Čech complex is denoted H^{∗}(U; F). If U ⊂ U′, then the natural map

H^{∗}(U) → H^{∗}(U′)

is an isomorphism.

Now suppose that the full subcategory C′ generated by U happens to be stable under formation of products. Then we can compare the Čech complex of U with the cosimplicial replacement of the functor F, which is defined to be the complex

∏_{U_{0}} F U_{0} ⇉ ∏_{U_{0}←U_{1}} F U_{1} ….

The natural map goes from the Čech complex to the cosimplicial replacement, and is in degree n, the map whose component with index U_{0} ← ⋯ ← U_{n} is the projection mapping to the factor with index (U_{0}, …, U_{n}).

Proposition 6.5. With the above assumptions, the natural map from the Čech complex to the cosimplicial replacement is an isomorphism.

Proof: The natural map is clearly (?) an isomorphism of H^{0}. Since the Čech cohomology groups take a short exace sequence of presheaves to a long exact sequence in cohomology, it suffices to show that the Čech cohomology groups are effaceable. For this it is enough to show that the groups H^{i}(U; F), i > 0, are zero when F is the right Kan extension of functor which assigns to a fixed object V ∈ U an abelian group A. More explicitly, the presheaf F is given by

F U = ∏_{V →U} A.

Now the let S be the set {f : V → U | U ∈ U}. Then the Čech complex of F is just the complex

A^{S} ⇉ A^{S×S} ….

For any set S, the simplicial set ES = S ⇇ S × … is contractible. It follows that the groups H^{i}(U; F) are the cohomology groups H^{i}(ES; A) which are zero for i > 0. This completes the proof.

6.3. The quasi-etale topology. Now let \underline{Aff} be the opposite of the category of commutative rings. There are several topologies we need to consider. A collection {U_{i} → U} is a covering in the flat topology if the map ∐ U_{i} → U is faithfully flat. This implies, in particular, that each U_{i} → U is flat. The fppf topology is obtained by requiring that the collection {U_{i} → U} be finite, and that each U_{i} → U be finitely presented. The quasi-etale topology is obtained by requiring that the complexes Λ_{U_{i}/U} be acyclic.

I guess I should make the following definition:

Definition 6.6. A map R → S of rings is quasi-unramified if the complex of S-modules Λ_{S/R} is acyclic. A map is quasi-etale if it is quasi-unramified and and flat.

ELLIPTIC COHOMOLOGY (February 22, 1999) 25

6.4. The associated stack. Now suppose that C is a Grothendieck topology and F is a contravariant functor from C to groupoids. The point of this section is to associate a stack to F.

Let J = {U_{i} → U} be a covering in C. A descent datum for F relative to J consists of a collection of objects x_{i} ∈ ob F (U_{i}), together with isomorphisms t_{i,j} : π_{i}^{∗}x_{i} → π_{j}^{∗}x_{j} satisfying the cocycle condition

π_{i}^{∗}t_{j,k} ∘ π_{k}^{∗}t_{i,j} = π_{j}^{∗}t_{i,k}.

For the covering J, define

Ũ_{0} = ∐ U_{i}
Ũ_{1} = ∐ U_{i} ×_{Y} U_{j}.

Then J̃ = (Ũ_{0}, Ũ_{1}) is naturally a groupoid object of C, which we may regard as a contravariant functor from C to groupoids. The category of descent data for F relative to J is the category of morphisms of groupoids from J̃ to F.

6.5. The cotangent complex of a stack. In this section we will consider the cotangent complex of a morphism M → N of stacks. This object will be a chain complex of quasicoherent O_{M}-modules, and is determined by insisting on the transitivity and flat base change properties. Unlike in the case of rings, it will not be the case that H_{i}L_{M/N} = 0 if i < 0.

By the transitivity sequence, the complex L_{M/N} will be the cofiber of the map L_{M} → L_{N}, and so we need only define the absolute complex. We will consider the case of a stack of the form M(A, Γ).

We will need a lemma. For a Hopf-algebroid (A, Γ) let B(A, Γ)^{•} be the cosimplicial object which represents the nerve of the groupoid represented by (A, Γ). One has

B(A, Γ)^{n} = Γ^{⊗n}

where ⊗ means “tensor over A,” and Γ^{⊗0} = A. Given an (A, Γ) co-module M, let B(M)^{•} be cosimplicial module over B(A, Γ) with B(M)^{n} = Γ^{⊗n} ⊗ M.

Lemma 6.7. The functor B from (A, Γ) co-modules to cosimplicial B(A, Γ)^{•} modules has a right adjoint Rcom.

Proof: I think this can be done with the Freyd theorem, but that approach isn’t very illuminating. I haven’t figured out the explicit formula yet.

Working remark 10. There is another notation which is helpful useful. Suppose we are given a stack M whose diagonal is representable, and a cover f : spec A → M. Write

spec Γ ∼ spec A ×_{M} spec A.

Then (A, Γ) is a Hopf-algebroid. Set f_{•} : spec B(A, Γ) → M be the augmentation. Strictly speaking, this map does not exist, but if one first converts the map f into a “fibration,” then one can find an equivalent simplicial stack, and a map to M. If we identify (A, Γ) co-modules with quasi-coherent M-modules, then the functor denoted B above would naturally be denoted f_{•}^{∗}. It’s right adjoint is then (f_{•})_{∗}.

I guess that we need to make use of the functorial L, and define

L_{M(A,Γ)} := Rcom L_{B(A,Γ)}.

I’ll have to verify some of the basic properties.

Proposition 6.8. If the 1-morphism M → N is representable, then the sheaves H_{i}L_{M/N} are zero for i < 0.

Proof: Choose a faithfull flat cover f : spec A → N and consider the 2-category pullback

spec B ⟶^{g} M
↓  ↓
spec A ⟶_{f} N .

Then by flat base change, g^{∗}L_{M/N} ∼ L_{B/A}. Since f is faithfully flat, so is g, and so

H_{i}L_{M/N} = 0 ⇔ g^{∗}H_{i}L_{M/N} = 0

⇔ H_{i}g^{∗}L_{M/N} = 0 ⇔ H_{i}L_{B/A} = 0.

But H_{i}L_{B/A} = 0 for i < 0.

ELLIPTIC COHOMOLOGY (February 22, 1999) 27

7. The moduli stack of formal groups

The point of this section is to establish some basic notation, and give the Landweber criteria for a map to be flat.

So, among other things, we have to define M_{FG}^{[n]} to be the moduli stack of formal groups of height greater than or equal to n. Over M_{FG}^{[n]} there is a global section v_{n} of ω^{p^{n}−1}. The zero set of v_{n} is M_{FG}^{[n+1]}, and it’s complement is M_{FG}^{(n)}. The stack M_{FG}^{(n)} is the moduli stack over ℤ/p of formal group laws, such that locally, in a local coordinate, “multiplication by p” is represented by a power series of the form λx^{p^{n}} + · · ·.

Working remark 11. The moduli stack of formal group laws, all of whose geometric fibers have height greater than or equal to n does not come from a Hopf algebroid. It is a colimit of ones that do, though. It is the colimit of (spec L/I_{n}^{k}, spec W/I_{n}^{k}).

7.1. The Frobenius morphism. Now let G → S be a formal group over a scheme S, which is, itself a scheme over spec Z/p. The morphism on rings which raises each element to the p^{th} power gives a commutative diagram

\begin{matrix}G & ⟶^{ϕ} & G \\ ↓ & & ↓ \\ S & ⟶_{ϕ} & S, \end{matrix}

and hence a map f_{G} : G → ϕ^{∗}G. If x is a local coordinate on G, then f_{G}(x) = x^{p}.

Write ω for the invariant differential on G. In a local coordinate x write “multiplication by p” as [p](x). Since p^{∗}ω = pω = 0 it follows that [p]′(x) = 0, and hence that [p](x) = g(x^{p}) for some power series g. This means that we can write p = v_{G}f_{G} for some v_{G} : ϕ^{∗}G → G. Since p is surjective, the map v_{G} is uniquely determined by this equation.

Now suppose that the height of G is ≥ n, with n > 1. Then, in a local coordinate, [p](x) = cx^{p^{n}} + … for some c. It follows that v_{G}(x) = cx^{p^{n−1}} + …, and that v′_{G}(0) = 0. But this means that v_{G}^{∗}ω = 0 and hence that v′_{G}(x) = 0, and so we can write v_{G}(x) = v_{G}^{(2)}(x^{p}) for some h. Writing this globally we see that there is a unique map v_{G}^{(2)} : (ϕ^{2})^{∗}G → G with v_{G}^{(2)} ∘ f_{G}^{ϕ} ∘ f_{G}. Iterating this discussion leads to the following result.

Proposition 7.1. If G has height ≥ n then there is a unique map v_{G}^{(n)} : (ϕ^{n})^{∗}G → G with the property that

v_{G}^{(n)}f_{G}^{ϕ^{n−1}} ⋯ f_{G} = p : G → G.

If the height of G is equal to n, then the map v_{G}^{(n)} is a isomorphism.

Proposition 7.2. For n ≥ 1, the 1-morphism M_{FG}^{(n)} → spec ℤ/p is quasi-etale.

Proof: The map is flat, so it suffices to show that it is quasiunramified. We will use the criteria of Proposition 5.12. We need to show that ϕ^{n} : M_{FG}^{(n)} → M_{FG}^{(n)} is an equivalence. But this is the 1-morphism sending a formal group G to the formal group (ϕ^{n})^{∗}G. The map v_{G}^{(n)} defines a natural isomorphism from this functor to the identity functor.

ELLIPTIC COHOMOLOGY (February 22, 1999) 29

8. Elliptic curves

8.1. The moduli stack. I’ll do better with the geometric motivation later. For now, let’s define a contravariant groupoid valued functor on \underline{Aff} by taking the objects to be the set of projective plane curves given by a Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

which we will also write in the form F (x, y) = 0, where

F (x, y) = y^{2} + a_{1}xy + a_{3}y − (x^{3} + a_{2}x^{2} + a_{4}x + a_{6}).

A morphism from the curve defined by F_{1}(x, y) = 0 to the curve defined by F_{2}(x, y) = 0 is a transformation

\begin{matrix}x & ↦ λ^{2}(x + r) \\ y & ↦ λ^{3}(y + sx + t) \end{matrix}  λ a unit

sending a solution of F_{1}(x, y) = 0 to a solution to F_{2}(x, y) = 0. If (x, y) is a solution of F (x, y) = 0, then the point

(λ^{2}(x + r), λ^{3}(y + sx + t))

is a solution to

λ^{6}F (λ^{−2}x − r, λ^{−3}y − s(λ^{−2}x − r) − t) ,

which can be rewritten as

y^{2} + η_{R}(a_{1})xy + η_{R}(a_{3})y = x^{3} + η_{R}(a_{2})x^{2} + a_{4}x + η_{R}(a_{6}),

where

(8.1) y^{2} + l(a_{1} − 2s)xy + l^{3}(a_{3} − ra_{1} − 2t − 2rs)y =

x^{3}+l^{2}(a_{2}+sa_{1}−s^{2}−3r)x^{2}+l^{4}(a_{4}+sa_{3}−2ra_{2}+(t−2rs)a_{1}−2st+2rs^{2}+3r^{2})x+

l^{6}(a_{6}−a_{4}r+a_{2}r^{2}−r^{3}−a_{3}rs+a_{1}r^{2}s−r^{2}s^{2}+a_{3}t−a_{1}rt+2rst−t^{2})

Working remark 12. OK. Some notation. f_{X} = f_{X/S} is the relative Frobenius.

8.2. Height and the Frobenius isogeny. Let A be a p-divisible group over a base S, and

f = f_{A/S} : A → ϕ^{∗}A

the relative Frobenius map. Since multiplication by p is is surjective and kills the kernel of f, there is a unique map

v = v_{A/S} : ϕ^{∗}A → A,

satisfying vf = p. Since fvf = pf = fp, and since f is surjective, it also follows that vf = p.

Let E be a generalized elliptic curve over a base S, E^{f} its formal completion, and i : E^{f} → E the canonical map. The map i gives rise to an isomorphism ker(f_{E^{f}}) → ker(f_{E}). When S is the spectrum of a field, and E is supersingular, the map i restricts to an isomorphism between the kernels of multiplication by p. This implies that the map ker(v_{E^{f}}) → ker(v_{E}) induced by i, is also an isomorphism. Finally, it follows for general S, that if all of the geometric fibers of E are supersingular, then the map ker(v_{E^{f}}) → ker(v_{E}) is an isomorphism. This is proved by reducing to the case where S is Noetherian and affine (for example, the universal case) and then using the fact that a map of finitely generated modules over a Noetherian ring is an isomorphism if and only if it is over every geometric fiber. I think there is an easier approach, since what really needs to be shown is that the kernel of mult by p is infinitesimal.

8.3. The invariant differential. Let E be the elliptic curve given by the Weierstrass equation

F (x, y) = y^{2} + a_{1}xy + a_{3} − (x^{3} + a_{2}x^{2} + a_{4}x + a_{6}) = 0. (8.2)

Introduce the coordinates z = −x/y, w = −1/y. Then z is a local parameter near the identity section (the point at infinity). The invariant differential is given by

ω = \frac{dx}{F_{y}} = −\frac{dy}{F_{x}} .

We need to expand ω as a power series in z.

ELLIPTIC COHOMOLOGY (February 22, 1999) 31

In terms of w and z, the invariant differential ω is

ω = \frac{w dz}{(3z^{3} − 2w) + 2a_{1}wz + 2a_{2}wz^{2} + a_{3}w^{2} + a_{4}w^{2}z}

= \frac{dz}{(3z^{3}/w − 2) + 2a_{1}z + 2a_{2}z^{2} + a_{3}w + a_{4}wz} ,

and the equation (8.2) becomes

w = z^{3} + a_{1}wz + a_{2}wz^{2} + a_{3}w^{2} + a_{4}w^{2}z + a_{6}w^{3}. (8.3)

It is also useful to rewrite this equation as

\frac{z^{3}}{w} = 1 − a_{1}z − a_{2}z^{2} − a_{3}w − a_{4}wz − a_{6}w^{2}. (8.4)

From these two equations one concludes

w = z^{3} + O(z^{4})
\frac{z^{3}}{w} = 1 − a_{1}z − a_{2}z^{2} − a_{3}z^{3} + O(z^{4}),

and finally that

ω = \frac{dz}{1 − a_{1}z − a_{2}z^{3} − 2a_{3}z^{3} + O(z^{4})}.

Lemma 8.5. Suppose that E is an elliptic curve over a ring R with identity section e, and

u ∈ H^{0}(O(−e)/O(−5e))

is a local parameter at e modulo degree 5. There are unique sections x ∈ H^{0}(O(2e)) and y ∈ H^{0}(O(3e)) satisfying

(8.6)  x ≡ u^{−2} mod O(e)
(8.7)  y ≡ u^{−3} mod O(2e)
(8.8)  x/y ≡ u mod O(5e).

Proof: The question is local in R so we may suppose that E is given by a Weierstrass equation

y^{2} + a_{1}x + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

with x and y satisfying (8.6) and (8.7). For convenience write z = x/y. Then z is a local parameter near e, and z ≡ u modulo O(2e). Write

z ≡ u + cu^{2} + O(u)^{3},

and consider a change of variables of the form y ↦ y + sx. Then

z ↦ z − sz^{2} + O(z)^{3},

and so there is a choice of x and y with the property that x/y ≡ u modulo O(3e). This choice of x and y is unique up to a change of variables of the form

\begin{matrix}x & ↦ x + r \\ y & ↦ y + t. \end{matrix}

Now write z ≡ u + cu^{3} + O(u^{4}). Replacing x by x + r replaces z with

z + r \frac{1}{y} ≡ z + ru^{3} + O(u^{4}).

It follows that x and y can be chosen so that x/y ≡ u modulo O(4e), and that this choice of x and y is unique up to a change of variables of the form

y ↦ y + t.

Finally, write z = u+cu^{4} +O(u^{5}). The transformation y ↦ y+t sends z to

\frac{x}{y + t} = \frac{x}{y(1 + \frac{t}{y})} ≡ z − tz^{4} + O(z^{5}).

It follows that x and y can be chosen so that z/y ≡ u modulo O(−5e), and that this choice is unique.

Corollary 8.9. The ring A = ℤ[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}] represents the functor which “elliptic curves together with a local parameter at e modulo degree 5.” The Hopf-algebroid (A, Γ) can be identified with the one representing the functor “elliptic curves with a local parameter at e modulo degree 5, and changes of local parameter.

8.4. Representability of M_{Ell} → M_{FG}. The point of this section is to give a proof of the following result.

Proposition 8.10. The 1-morphism M_{Ell} → M_{FG} is representable.

The proof makes use of an auxilliary stack.

Definition 8.11. A formal group law chunk modulo degree n over a ring R is an element

F (x, y) ∈ R[x, y]/(x, y)^{n}

ELLIPTIC COHOMOLOGY (February 22, 1999) 33

satisfying

F (x, y) = F (y, x)
F (x, 0) = F (0, x) = x
F (F (x, y), z) = F (x, F (y, x)).

A homomorphism f : F_{1} → F_{2} of formal group law chunks is an element f ∈ R[t]/(t)^{n} with the property that

f(F_{1}(x, y)) = F_{2}(f(x), f(y)).

The functor “formal group law chunks over R modulo degree n and isomorphisms” is represented by a Hopf-algebroid. The theory is much the same as in the case of formal group laws. The ring representing the functor “set of formal group law chunks modulo degree n + 2” is isomorphic to

ℤ[x_{1}, …, x_{n}],

and the universal “group chunk” is given, modulo products of the x_{i}’s, by

x + y + ∑ x_{n}c_{n}(x, y),

where c_{n}(x, y) = d_{n}((x + y)^{n} − x^{n} − y^{n}), and

d_{n} = \begin{cases}\frac{1}{p} & if n = p^{j}, p a prime \\ 1 & otherwise. \end{cases}

The universal isomorphism between group chunks is given by the change of variables

x ↦ t_{0}x + ⋯ + t_{n}x^{n+1} + O(x^{n+2}),

with t_{0} a unit.

The stack associated to the Hopf algebroid of formal group law chunks modulo degree n will be denoted M_{FG(n)}. A 1-morphism spec R → M_{FG(n)} will be called a formal group chunk modulo degree n over R. By construction, there are 1-morphisms

⋯ → M_{FG(n+1)} → M_{FG(n)} → …,

and the “homotopy inverse limit” of this tower is just M_{FG}. Since they come from Hopf algebroids, the diagonal map of each M_{FG(n)} is representable.

We will be interested in the case n = 5. In this case the universal group law chunk can be written as

F (x, y) = x + y + x_{1}c_{2} + x_{2}c_{3} + (x_{3}c_{4} + x_{2}x_{1}c_{2}^{2}) + O(x, y)^{5}.

The universal invariant differential is

\frac{dx}{F_{y}(x, 0)} = \frac{dx}{1 + x_{1}x + x_{2}x^{2} + 2x_{3}x^{3} + O(x)^{4}}

= \begin{aligned}(1 − x_{1}x & + (x_{1}^{2} − x_{2})x^{2} \\ & + (−x_{1}^{3} + 2x_{1}x_{2} − 2x_{3})x^{3} + O(x)^{4}) dx. \end{aligned}

Lemma 8.12. The 2-category pulback of

M_{Ell}

↓

spec ℤ[x_{1}, x_{2}, x_{3}] ⟶ M_{FG(5)}

is representable. It is represented by the elliptic curve

y^{2} + a_{2}x + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

over the ring A with local parameter t = x/y modulo t^{5}.

Proof: The pullback stack is the stack which associates to each ring R an elliptic curve over R, a formal group law chunk over R modulo degree 5, together with an isomorphism between the associated formal group chunk and the formal completion of the elliptic curve modulo degree 5. This data is clearly equivalent to giving an elliptic curve over R and together with a local parameter at e modulo degree 5. The result follows from Corollary 8.9

Working remark 13. Don’t forget that you are really working with a scheme, and not over the ring A. Unless you first set things up so that you allow “additive” reduction (this might be a good idea), and then remove this fiber when you relate things to homotopy theory. Say, why not do it like this. Leave in the “bad point” until it comes time to verify flatness of the map to the stack of formal groups. That map is, of course, not flat unless you remove the bad fiber.

ELLIPTIC COHOMOLOGY (February 22, 1999) 35

8.5. Some vanishing results. I’m going to write as if I have defined certain objects.

Fix a prime p, and let M_{Ell,p}^{ord} (respectively M_{Ell,p}^{ss}) be the moduli stacks of ordinary (respectively supersingular) elliptic curves in charasteric p. There are therefore 2-cartesian squares

M_{Ell,p}^{ord} ⟶ M_{Ell}
↓  ↓
M_{FG}^{(1)} ⟶ M_{FG}

and

M_{Ell,p}^{ss}  M_{Ell,p}^{ss} ⟶ M_{Ell}
↓  ↓  ↓
M_{FG}^{(2)} ⟶ M_{FG}^{[2]} ⟶ M_{FG}

Proposition 8.13. The 1-morphism M_{Ell,p}^{ss} → M_{FG}^{(2)} is quasietale.

Proof: Being the base change of a flat map it is flat, so it suffices to show that it quasi-unramified. Since M_{FG}^{(2)} → spec ℤ/p is quasi-unramified, it suffices, by the transitivity sequence to show that M_{Ell,p}^{ss} → spec ℤ/p is quasi-unramified. The proof of this is similar to that of Proposition 7.2, and uses the fact that for an supersingular elliptic curve A, the map v_{A}^{(2)} : (ϕ^{2})^{∗}A → A is an isomorphism.

8.6. A quasi-etale cover. The point of this section is to exhibit a quasi-etale cover of M_{Ell}. This is needed in order to know that the cohomology of the Čech complexes we are computing actually give the cohomology of the whole stack. It is also useful in identifying the cotangent complex of ℓ with ω^{10}, since, among other things, it shows that the cotangent complex has all of its homology in dimension 0. The cover will be exhibited in pieces. We will consider separately the cases when 2 is inverted, and when 3 is inverted. Over each of these opens we will consider separately the case when Δ is inverted, and when c_{4} is inverted.

We will need a couple of algebraic lemmas.

Let R be a ring. For an R-module M, let M[x] denote the Rmodule M ⊗_{R} R[x], and write elements of M[x] as polynomials with coefficients in M.

Lemma 8.14. Let f(x) = a_{n}x^{n} + … a_{0} be an element of R[x]. If the ideal generated by the a_{i} is the unit ideal, then f does not annihilate any non-zero element of M[x].

Proof: Suppose that the ideal (a_{0}, …, a_{n}) is the unit ideal and let

g(x) = m_{k}x^{k} + ⋯ + m_{0} ∈ M[x]

be an element of minimal degree which is annihilated by f. Let j ≥ 0 be the smallest number with the property that the elements a_{n}, …, a_{j} annihilate g. If j > 0 then

f(x)g(x) = a_{j−1}m_{k}x^{k+j−1} + lower terms.

It follows that a_{j−1}m_{k} = 0, and so a_{j−1}g(x) has lower degree than g and is annihilated by f, and so must be zero. It follows that j = 0 and that g is annihilated by all of the a_{i}. But this implies that g = 0 since the a_{i} generate the unit ideal.

Lemma 8.15. Suppose that R is a ring, and that

f(x) = a_{n}x^{n} + ⋯ + a_{0} ∈ R[x]

is a polynomial. The module R[x]/(f(x)) is flat if and only if the ideal (a_{0}, …, a_{n}) is the unit ideal.

Proof: I’ll just do the “if” part, so suppose that the a_{i} generate the unit ideal. Then Lemma 8.15 with M = R shows that the sequence

R[x] ⟶^{·f} R[x] → R[x]/(f)

is short exact, and hence a free resolution. Tensoring with an arbitrary R-module M and once again using Lemma 8.15 gives the result.

Corollary 8.16. Suppose that R is a ring and that f ∈ R[x] is not a divisor of zero. If the ideal (f, f′) is the unit ideal then B = R[x]/(f) is quasi-etale over R.

Proof: The fact that (f, f′) is the unit ideal implies that the coefficients of f generate the unit ideal. It follows from Lemma 8.15 that R[x]/(f) is flat over R and from Lemma 8.14 that f is not a divisor of zero. The Meyer-Vietoris sequence (Corollary 4.2) then shows that the complex L_{B/R} is quasi-isomorphic to

B ⟶^{·f′} B

ELLIPTIC COHOMOLOGY (February 22, 1999) 37

which is acyclic, since f′ is a unit in B.

The following is a useful special case is.

Corollary 8.17. Suppose R is a ring, n is an integer, and that a and n · 1 are units in R. Then the map R → R[x]/(x^{n} − a) is quasi-etale.

8.6.1. When 2 and Δ are units. Consider the elliptic curve

y^{2} = x^{3} + τx^{2} + x.

over the ring R = ℤ[\frac{1}{2}][τ, (τ^{2} − 4)^{−1}]. With 2 inverted, the stack M_{Ell} (oops) is presented by the Hopf algebroid (B, Γ_{B}), with B = ℤ[\frac{1}{2}][b_{2}, b_{4}, b_{6}], Γ = B[r, λ^{±1}]. The map spec B → M_{Ell} is the one classifying the elliptic curve

y^{2} = x^{3} + b_{2}x^{2} + b_{4}x + b_{6} = f(x),

and the universal isomorphism is given by the transformation

x ↦ λ^{−2}x + r.

Replacing x with x + r changes b_{6} to f(r), and b_{4} to f′(r). Replacing x with λ^{−2}x sends b_{4} to λ^{4}b_{4}. The 2-category fiber product of spec R and spec B over M_{Ell} is therefore represented by B[t, α]/(f(t), α^{4} − f′(t)). The ring B[t]/f(t) is etale over B because the discriminant of f is a unit. Over B[t]/f(t), the element f′(t) is a unit, and so the addition of α defines a further etale extension.

8.6.2. When 2 and c_{4} are units. Consider the elliptic curve

y^{2} = x^{3} + x^{2} + τ.

over the ring R = ℤ[\frac{1}{2}][τ, (τ^{2} − 4)^{−1}]. The value of c_{4} is 16 for this curve, and so is a unit since 2 has been inverted. The value of c_{4} for the universal curve is 16(b_{2}^{2} − 3b_{4}). Making the change of variables x ↦ x + r changes b_{4} to f′(r), and sends b_{2} to b_{2} + 3r. The 2-category fiber product is therefore represented by

B[t, α]/(f′(t), α^{2} − (b_{2} + 3t)).

Adjoining t is flat, since the coefficients of f′ generate the unit ideal in R. It is unramified, since the discriminant of f′ is 4(b_{2}^{2} − 3b_{4}) = \frac{c_{4}}{4} , which is a unit. The element b_{2} + 3t is a unit since it is \frac{1}{2}f′′(t), so adjoining α is etale.

8.6.3. When 3 and Δ are units. Now consider the curve

y^{2} + τxy + y = x^{3}.

The discriminant is τ^{3} − 27, so it is non-singular over the ring

R = ℤ[\frac{1}{3}][τ, (τ^{3} − 27)^{−1}].

8.6.4. When 3 and c_{4} are units. Now consider the curve

y^{2} + xy = x^{3} + τ. (8.18)

The value of c_{4} is 1 and the value of c_{6} is 1 + 864τ. It follows that the value of Δ is −τ(1 + 432τ). We will consider this curve over the ring

R = ℤ[\frac{1}{3}][τ].

To verify that it is quasi-etale over M_{Ell} it suffices, since M_{Ell} is flat over ℤ, to verify that it is quasi-etale after reducing mod 2, and after inverting 2.

Modulo 2, the value of c_{4} = a_{1}^{4} is a unit. There is a unique transformation x ↦ x + r (namely r = −a_{3}/a_{1}) which gets rid of the term a_{3}y. There is a unique transformation y ↦ y + t (namely t = a_{4}/a_{1}) which gets rid of the term a_{4}x^{2}. Finally, there is a unique value of λ which brings a_{1} to 1. This shows that, modulo 2 and with c_{4} inverted, the stack we are interested is associated to the curve

y^{2} + xy = x^{3} + α_{2}x^{2} + α_{6},

and the transformation y ↦ y + sx. Making the substitution y ↦ y + sx leaves α_{6} unchanged, and sends α_{2} to s^{2} + s + α_{2}. It follows that the 2-category pullback

spec ℤ/2[α_{2}, α_{6}]
↓
spec ℤ/2[d] ⟶  M_{Ell}

is spec ℤ/2[α_{2}, α_{6}][s]/(f(s)), where f(s) = s^{2}+s+α_{2}. Since f′(s) = 1 the map is quasi-etale by Corollary 8.16.

Now suppose that 2 has been inverted, so that, in fact, 6 is a unit. We have seen (have we?) that the map

spec ℤ[\frac{1}{6}][a_{6}] → M_{Ell}[\frac{1}{6}]

ELLIPTIC COHOMOLOGY (February 22, 1999) 39

classifying the curve

y^{2} = x^{3} + x + a_{6} (8.19)

is a faithfully flat cover of the locus where c_{4} is a unit. To compute the 2-category pullback of

spec ℤ[\frac{1}{6}][a_{6}]
↓
spec ℤ[\frac{1}{6}][τ] ⟶ c_{4}^{−1}M_{Ell}[\frac{1}{6}]

first note that in order that a transformation

x ↦ l^{−2}(x + r)
y ↦ l^{−3}(y + sx + t)

send an equation of the form (8.19) into one of the form (8.18) one must have

\begin{aligned}t & = 0 \\ 3r & = s^{3} \end{aligned}  \begin{aligned}s^{4} & = −3 \\ l & = 1/2s. \end{aligned}

For the record, the value of τ is then (3s^{2}a_{6} −2)/1728. This means that the 2-category pullback is given by spec ℤ[\frac{1}{6}][a_{6}][s]/(s^{4} + 3). This is quasi-etale over ℤ[\frac{1}{6}][a_{6}] by 8.17.

Working remark 14. Better name the stacks Δ^{−1}M_{Ell} and c_{4}^{−1}M_{Ell}.

9. Determination of certain cotangent complexes

9.1. The elliptic moduli stack. In this section let M be the stack M_{Ell}.

Proposition 9.1. The natural map (?) L_{M_{Ell}} → Ω_{M_{Ell}}^{1} is a quasiisomorphism.

Proof: This is immediate from the fact that the stack M_{Ell} admits a quasi-etale cover by smooth algebras.

Proposition 9.2. There is a unique 1 form ν on M_{Ell} with

1728 · ν = 2c_{4} dc_{6} − 3c_{6} dc_{4}.

The form ν gives an isomorphism

ω^{±10} ≈ Ω_{M_{Ell}}^{1}.

Proof: It suffices to check this locally on the cover given in section 8.6. For the curve

y^{2} = x^{3} + τx^{2} + x

over ℤ[\frac{1}{2}][τ, (τ^{2} − 4)^{−1}], the values of c_{4} and c_{6} are given by

c_{4} = 16τ^{2} − 48
c_{6} = 64τ^{3} − 288τ

and so ν = 16 dτ exists, is unique, and is a nowhere vanishing section of the cotangent bundle.

For the curve

y^{2} = x^{3} + x^{2} + τ,

over ℤ[\frac{1}{2}][τ], the values of c_{4} and c_{6} are given by

c_{4} = 16
c_{6} = 32(27τ + 2),

and so ν = 16 dτ exists, is unique, and is a nowhere vanishing section of the cotangent bundle.

For the curve

y^{2} + τxy + y = x^{3}

ELLIPTIC COHOMOLOGY (February 22, 1999) 41

over ℤ[\frac{1}{3}][τ, (τ^{3} − 27)^{−1}], the values of c_{4} and c_{6} are given by

c_{4} = τ^{4} − 24τ
c_{6} = τ^{6} − 36τ^{3} + 216

and so ν = 9 dτ exists, is unique, and is a nowhere vanishing section of the cotangent bundle.

Finally, for the curve

y^{2} + xy = x^{3} + τ

over ℤ[\frac{1}{3}][τ], the values of c_{4} and c_{6} are given by

c_{4} = 1
c_{6} = 864τ + 1

and so ν = dr exists, is unique, and is a nowhere vanishing section of the cotangent bundle.

Corollary 9.3. The complex L_{M_{Ell,p}^{ord}} is quasi-isomorphic to ω^{±10}.

Proof: We know that the assertion is true of M_{Ell,p}, by flat base change. It then follows for M_{Ell,p}^{ord} since the inclusion M_{Ell,p}^{ord} → M_{Ell,p} is quasi-etale.

9.2. The cotangent complex of M_{Ell} over M_{FG}. To make the notation less cluttered, for this section write M = M_{Ell} and N = M_{FG}. The point of this section is to identify L_{M/N} ⊗ ℤ/p and L_{M/N} /otimesℚ.

Proposition 9.4. For each prime p, the complex L_{M/N} ⊗ ℤ/p is quasi-isomorphic to i_{∗}ω^{±10}, where i : M_{Ell,p}^{ord} → M is the inclusion.

Proof: It is useful to write M_{p} = M ⊗ ℤ/p, N_{p} = N ⊗ ℤ/p and look at the diagrams

M_{Ell,p}^{ss} ⟶^{k} M_{p} ⟵^{l} M_{Ell,p}^{ord}
↓  ↓  ↓
M_{FG}^{(2)} ⟶ N_{p} ⟵ M_{FG}^{(1)},

and

M_{p} ⟶^{j} M
↓  ↓
N_{p} ⟶ N .

in which the squares are 2-cartesian, and all of the vertical maps are flat. In this notation, the map i is the composite jl. Now, by definition, L_{M/N} ⊗ ℤ/p ∼ j_{∗}j^{∗}L_{M/N}. By flat base change, j^{∗}L_{M/N} ∼ L_{M_{p}/N_{p}}. Again, by flat base change, we have a weak equivalence

k^{∗}L_{M_{p}/N_{p}} → L_{M_{Ell,p}^{ss}/M_{FG}^{(2)}}.

But this latter complex is acyclic, by Proposition 7.2. Since k is locally defined by the vanishing of a single function which is not a divisor of zero, the means that the map

L_{M_{p}/N_{p}} → l_{∗}l^{∗}L_{M_{p}/N_{p}}

is a weak equivalence. By flat base change, the map

l^{∗}L_{M_{p}/N_{p}} → L_{M_{Ell,p}^{ord}/M_{FG}^{(1)}}

is a weak eqivalence. By Proposition 7.2 the map M_{FG}^{(1)} → spec ℤ/p is quasi-etale. It follows from the transitivity sequence that

L_{M_{Ell,p}^{ord}} → L_{M_{Ell,p}^{ord}/M_{FG}^{(1)}}

is an equivalence. But L_{M_{Ell,p}^{ord}} is quasi-isomorphic to ω^{10} by Corollary 9.3. Collecting these isomorphisms completes the proof.

Proposition 9.5. There is a natural quasi-isomorphism

L_{M/N} ⊗ ℚ ∼ (ω^{4} ⊕ ω^{6}) ⊗ ℚ.

Proof:

ELLIPTIC COHOMOLOGY (February 22, 1999) 43

10. Elliptic spectra

1. A = any elliptic curve

2. ω = usual line bundle

Definition 10.1. An elliptic spectrum E is etale if the map

spec π_{0}E → M_{Ell}

is etale open of finite type.

For a map U → M_{Ell}, define a graded ring E_{U}_{∗} by

E_{U}_{2k} = Γ_{U}(ω^{k})
E_{U}_{2k+0} = 0

Lemma 10.2. If U is affine, U → M is an etale open, and ω_{U} admits a nowhere vanishing global section, then the pair (E_{U}_{∗}, A_{U}) satisfies the conditions of the Landweber exact functor theorem. It has a unique homotopy associative multiplication.

Proof: It suffices to check these after fpqc base change. Do it on the standard cover.

Working remark 15. Is it fpqc or fppf?

Thus the etale elliptic spectra are those of the form E_{U}.

11. Factoring and lifting

11.1. The section (M_{Ell})′_{et} → ho A_{∞}.

Proposition 11.1. The functor spec π_{0} : ho A_{∞} → (M_{Ell})′_{et} has a unique section up to vertical natural equivalence.

Working remark 16. Either arrange that the two categories have the same objects, or work out what you mean by “vertical.”

Lemma 11.2. For an object U → M_{Ell} of (M_{Ell})′_{et}, the space

A_{∞} {E_{U}}

is non-empty and connected.

Proof: The group

Der_{π_{0}E_{U}}^{2} (π_{0}E_{U} ∧ E_{U}, E_{U}^{0}S^{1})

is zero because E_{U}^{0}S^{1} = 0. According to Theorem 12.12 the groups

Der_{π_{0}E_{U}}^{s+1} (π_{0}E_{U} ∧ E_{U}, E_{U}^{0}S^{t})

are zero for s > 1. The result therefore follows from the spectral sequence of Proposition 2.5.

Lemma 11.3. Suppose that E and F are cofibrant-fibrant objects of A_{∞}^{Ell} with spec π_{0}E and spec π_{0}F corresponding to U → M_{Ell} and V → M_{Ell} respectively. Then

i) The map A_{∞}^{Ell}(E, F) → M_{Ell}(V, U) is surjective.

ii) Given f ∈ M_{Ell}(U, V) let

A_{∞}^{Ell}(E, F)_{f} = (spec π_{0})^{−1}{f}.

Then the set of components of A_{∞}^{Ell}(E, F)_{f} is a principal homogeneous space for

π_{0}A_{∞}^{Ell}(F, F)_{1} ≈ Der_{π_{0}F}^{2} (π_{0}F ∧ E, F^{0}S^{2})

≈ (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V} .

ELLIPTIC COHOMOLOGY (February 22, 1999) 45

Proof: This all follows from the spectral sequence of Proposition 2.6 and the computation of Theorem 12.12. Indeed for any choice of basepoint the spectral sequence collapses at E_{2} and π_{0} is given by the principal homogeneous space above.

Proof of Proposition 11.1: By Proposition A.6 the obstruction to the existence of a section lies in

H^{2}(M_{Ell}; (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V}) = 0.

This proves the existence of a section. Given that there is a section, Proposition A.6 then asserts that the set of vertical natural equivalence classes of sections is a principal homogeneous space for

H^{1}(M_{Ell}; (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V}) = 0.

This completes the proof.

Working remark 17. Mention that two objects of A_{∞}^{Ell} are isomorphic in ho A_{∞} if and only if their images in (M_{Ell})′_{et} are isomorphic. Somewhere say that an object or map of A_{∞}^{Ell} is of type “P,” if its image in image in A_{∞} has P, where P is one of cofibrant, cofibration, fibrant, fibration, weak equivalence, etc, and ho A_{∞} to be the category with objects the cofibrant-fibrant objects of A_{∞} and with ho A_{∞}^{Ell}(E, F) = π_{0}A_{∞}^{Ell}(E, F).

Working remark 18. Somewhere define the category A_{∞}^{Ell} to consist only of objects whose underlying A_{∞} ring is cofibrant-fibrant.

11.2. The lift to A_{∞}. Proposition 11.1 gives a unique (up to isomorphism) functor

D : (M_{Ell})′_{et} → ho A_{∞}.

We now turn to the problem of lifting it to a functor

D̃ : (M_{Ell})′_{et} → A_{∞}.

The functor D̃ will automatically factor through A_{∞}^{Ell} since the diagram

A_{∞}^{Ell} ⟶ A_{∞}
↓  ↓
ho A_{∞}^{Ell} ⟶ ho A_{∞}

is a pullback.

Proposition 11.4. The diagram D is a centric diagram: for each map f : V → U in (M_{Ell})′_{et}, the map

A_{∞}(DV, DV)_{1} → A_{∞}(DU, DV)_{f}

is a weak equivalence.

Proof: We may (and must) suppose that DU and DV are cofibrant-fibrant obects of A_{∞}. The homotopy groups of the two spaces in question can be calculated with the spectral sequence of Proposition 2.6. By Corollary 12.13, the map of E_{2}-terms is an isomorphism. The result follows.

The homotopy groups of the realization space r(D) can now be computed using the spectral sequence of Proposition 3.2.

Proposition 11.5. For each t, the functor

U ↦ π_{t} Bhaut_{1} DU

is a sheaf (of abelian groups) on (M_{Ell})′_{et}. There is, for n > 1, an isomorphism

π_{2n−1} Bhaut_{1} D( −) ≈ ω^{n} ⊗ (Ω^{1})^{−1} ⊗ ∏_{p} (O_{M_{Ell}})_{p} /O_{M_{Ell}},

and an exact sequence

0 → ω^{n} ⊗ M → π_{2n} Bhaut_{1} D( −) → ω^{n} ⊗ ∏_{p} (v^{−1}Ô_{V} /O_{V} )_{p} → 0,

where M lies in the (splittable) exact sequence

0 → (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V} → M → ∏_{p}(O_{V})_{p}/O_{V} → 0.

ELLIPTIC COHOMOLOGY (February 22, 1999) 47

Proposition 11.6. For s > 0, the groups

H^{s}(M_{Ell}; π_{2n−1} Bhaut_{1} D( −))

are 0, and the map

H^{s}(M_{Ell}; π_{2n} Bhaut_{1} D( −)) →

H^{s} (M_{Ell}; ω^{n} ⊗ (Ω^{1})^{−1} ⊗ (v^{−1}Ô_{V} /O_{V} )_{p}) .

is an isomorphism. The bigraded group H^{s}(M_{Ell}; π_{t} Bhaut_{1} D( −) is therefore isomorphic to the bigraded abelian group

a_{6} · (a_{1}^{−2}ℤ_{2}[[a_{1}^{2}]][a_{6}, η, a_{1}η]/a_{6}ℤ_{2}[a_{2}^{±1}, a_{6}, η])

in which

|a_{i}| = (0, 2i) |η| = (1, 2).

and in which, to simplify notation, the relation 2η = 0 is implied, but not mentioned.

These latter groups are determined in 13.10.

Working remark 19. Oops, not quite. Remember the cusps!

12. Computations of Quillen cohomology

Working remark 20. I am muddling the distinction between a sheaf and it’s global sections. This needs cleaning. Also, this needs to be rewritten with the changes of base built into the U and V, and not the moduli stacks themselves. This makes the notation simpler, and after all, is how things come up when we need them.

The main purpose of this section is to calculate

Der_{π_{0}F} (π_{0}F ∧ E, F^{0}S^{t}) = Rhom^{i}(Λ_{V ×_{M_{FGL}}U/V} , s_{∗}O_{V}),

with respect to a map f : E → F of elliptic spectra, with

spec π_{0}E ↔ U → M_{Ell}
spec π_{0}F ↔ V → M_{Ell}
f ↔ s : V → V ×_{M_{FGL}} U.

The main results are Corollaries 12.7, 12.10, and Theorem 12.12 below.

12.1. Computations modulo a prime power. Fix a prime p, and let M_{Ell,m} and M_{FGL,m} be the moduli stacks obtained from M_{Ell} and M_{FGL} by change of base to ℤ/(p^{m}).

Suppose given objects

U → M_{Ell,m}, V → M_{Ell,m} ∈ ob (M_{Ell,m})′_{et} ,

and a section s : V → V ×_{M_{Ell,m}} U. Let V^{ord} ⊆ V be the subscheme

of V defined by the condition that all of the geometric fibers of A_{V} are ordinary elliptic curves. Thus

V^{ord} = V ×_{M_{Ell,m}} M_{Ell,m}^{ord}

The complement of V^{ord} is V^{ss}. The closed subscheme V^{ss} is defined by the vanishing of a single function v ∈ O_{V}. In fact we can take v = u^{−p^{m−1}(p−1)}E_{p−1}^{p^{m−1}} (or even v = u^{1−p}E_{p−1}), where u is a nowhere vanishing section of ω, and E_{p−1} = v_{1} is the Eisenstein series. The function v is not a divisor of zero, by the assumptions we have made.

ELLIPTIC COHOMOLOGY (February 22, 1999) 49

Working remark 21. Notation: V → M_{Ell} classifies A_{V}. M_{Ell,m} is the reduction of M_{Ell} mod p^{m}. You might write

V ×_{M_{FGL,m}} U = Isom(π_{1}^{∗}A_{V}^{f} , π_{2}^{∗}A_{U}^{f}).

Proposition 12.1. The complex of sheaves Λ_{V ×_{M_{FGL,m}}U/V} is quasi-

isomorphic to the sheaf

i_{∗}π_{2}^{∗}Ω_{U}^{1} = (v × 1)^{−1}π_{2}^{∗}Ω_{U}^{1},

where

i : V^{ord} ×_{M_{FGL,m}} U → V ×_{M_{FGL,m}} U and
π_{2} : V^{ord} ×_{M_{FGL,m}} U → U

are the inclusion and projection respectively.

The proof depends on a couple of other results.

Lemma 12.2. i) The map

V^{ord} ×_{M_{FGL,m}} U → V^{ord} × U

is formally etale. It factors as a formally etale surjective map

π : V^{ord} ×_{M_{FGL,m}} U = V^{ord} ×_{M_{FGL,m}} U^{ord} → V^{ord} × U^{ord}

followed by a closed immersion.

ii) The map

V^{ss} ×_{M_{FGL,m}} U → V^{ss}

is formally etale.

proof of Proposition 12.1, given Lemma 12.2: Let

j : V^{ss} ×_{M_{FGL,m}} U → V ×_{M_{FGL,m}} U

be the inclusion. Since V ×_{M_{FGL,m}} U → V is flat, the map

j^{∗}Λ_{V ×_{M_{FGL,m}}U/V} → Λ_{V^{ss} ×_{M_{FGL,m}}U/V}

is a quasi-isomorphism (Lemma 5.2). By part ii) and an algebraic lemma I haven’t mentioned, the complex Λ_{V^{ss} ×_{M_{FGL,m}}U/V} is acyclic,

so by Lemma 5.5 the map

Λ_{V ×_{M_{FGL,m}}U/V} → i_{∗}Λ_{V^{ord} ×_{M_{FGL,m}}U/V} = i_{∗}Λ_{V^{ord} ×_{M_{FGL,m}}U/V^{ord}}

is an equivalence. By the transitivity sequence, and part i) of Lemma 12.2, the map

π^{∗}Λ_{V^{ord}×U/V^{ord}} → Λ_{V^{ord} ×_{M_{FGL,m}}U/V^{ord}}

is an equivalence. Finally, since U is smooth of dimension 1, the computation of Lemma 5.7 gives a weak equivalence

Λ_{V^{ord}×U/V^{ord}} ≈ O_{V^{ord}} ⊗ Ω_{U}^{1}.

The result now follows by assembling this chain of isomorphisms.

Lemma 12.3. Suppose that R is a ring, and v ∈ R an element which is not a divisor of zero (is this necessary). If P is a projective module and M is a module for which v : M → M is a monomorphism, then

Ext^{i}(v^{−1}P, M) = \begin{cases}hom_{R}(P, v^{−1}M̂/v^{−1}M) & i = 1 \\ 0 & i > 1, \end{cases}

where

M̂ = \underleftarrow{lim}_{k}M/v^{k}M.

Proof: The long exact sequence in Ext coming from

M ↣ v^{−1}M ↠ v^{−1}M/M

gives (with hom being hom of R-modules)

hom(v^{−1}P, M) → hom(v^{−1}P, v^{−1}M) →

hom(v^{−1}P, v^{−1}M/M) → Ext(v^{−1}P, M) → 0.

ELLIPTIC COHOMOLOGY (February 22, 1999) 51

Now there are isomorphisms

hom(v^{−1}P, v^{−1}M) ≈ hom(P, v^{−1}M) (12.4)

v^{−1}M̂ ⟶^{≈} \underleftarrow{lim}_{n}v^{−1}M/v^{n}M (12.5)

(12.6) hom(P, \underleftarrow{lim}_{n}v^{−1}M/v^{n}M) ⟶^{≈} hom(v^{−1}P, v^{−1}M/M),

and with respect to these the map

\begin{matrix}v^{−1}M & → hom(v^{−1}R, v^{−1}M/M) \\ m & ↦ f_{m} f_{m}(r) = rm \end{matrix}

corresponds to the natural map v^{−1}M → v^{−1}M̂.

This completes the proof, save verifying the above isomorphisms. The first and third are clear. To see that (12.5) is surjective, suppose that a = (a_{n}) ∈ \underleftarrow{lim}_{n}v^{−1}M/v^{n}M is a sequence of elements of v^{−1}M with

a_{n+1} ≡ a_{n} mod v^{n}M.

Then for some N, v^{N}a_{0} ∈ M, so for all n ≥ 0, v^{N}a_{n} ∈ M. It follows that v^{N}a ∈ M̂, and so a is in the image of v^{−1}M̂. To see that (12.5) is injective, note that the kernel of

v^{−1}M̂ → v^{−1}M/v^{n}M

is v^{n}M̂. It follows that the kernel of (12.6) is is ∩v^{n}M̂ = 0.

Working remark 22. There must be something easier to say.

Corollary 12.7. There is an isomorphism of quasi coherent sheaves

Rhom^{i}(Λ_{V ×_{M_{FGL,m}}U/V} , s_{∗}O_{V}) ≈

\begin{cases}v^{−1}Ô_{V} /v^{−1}O_{V} ⊗ (Ω_{V}^{1} )^{−1} & i = 1 \\ 0 & otherwise, \end{cases}

where Ô_{V} = \underleftarrow{lim} O_{V} /v^{n}O_{V}.

Proof: Write

Λ = Λ_{V ×_{M_{FGL,m}}U/V} .

Then

Rhom^{i}(Λ, s_{∗}O_{V}) ≈ Ext^{i}((v × 1)^{−1}π_{2}^{∗}Ω_{U}^{1}, s_{∗}O_{V})
≈ Ext_{V}^{i} ((v)^{−1}s^{∗}π_{2}^{∗}Ω_{U}^{1}, O_{V}).

Since π_{2} ∘ s is etale, the natural map s^{∗}π_{2}^{∗}Ω_{U}^{1} → Ω_{V}^{1} is an isomorphism. Since V is smooth of dimension 1, this latter sheaf is locally free of rank one, and so it’s module of global sections is projective. The result now follows from Lemma 12.3, with R = O_{V}, v = v, P the module of global sections of Ω_{V}^{1} , and M = O_{V}.

12.2. Computations over a ℚ-algebra. Now let M_{Ell,0} and M_{FGL,0} be the moduli stacks obtained from M_{Ell} and M_{FGL} by change of base to ℚ. Fix

U → M_{Ell,0}, V → M_{Ell,0} ∈ ob (M_{Ell,0})′_{et} ,

and a section s : V → V ×_{M_{Ell,0}} U as before. For ease of notation,

set

Λ = Λ_{V ×_{M_{Ell,0}}U/V}

Lemma 12.8. The scheme V ×_{M_{Ell,0}} U is a trivial G_{m}-torsor over

V × U. It is therefore non-canonically isomorphic to

V × U × G_{m}

and in particular is smooth over V of dimension 2.

Proof: Since we are over a Q-algebra, the formal groups A_{U}^{f} and A_{V}^{f} are isomorphic to the formal completions of the additive groups ω_{U} and ω_{V} respectively. It follows that

V ×_{M_{Ell,0}} U ≈ Isom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U})
≈ Isom(π_{1}^{∗}ω_{V}^{f} , π_{2}^{∗}ω_{U}^{f}),

and so it is a G_{m}-torsor over V × U. Since we have assumed that the bundles ω are trivializable, the bundles π_{1}^{∗}ω_{V} and π_{2}^{∗}ω_{U} are isomorphic, and so this torsor does admit a section.

ELLIPTIC COHOMOLOGY (February 22, 1999) 53

Corollary 12.9. The transitivity sequence

π^{∗}Ω_{V ×U/U}^{1} → H_{0}Λ → Ω_{V ×_{M_{FGL}}U/V ×U}^{1}

is short exact and splittable. It can be identified with

0 → π_{2}^{∗}Ω_{U}^{1} → H_{0}Λ → hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) → 0.

There is an isomorphism

H_{1}Λ = π_{2}^{∗}Ω_{U}^{1} ⊗ hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}).

The sheaves H_{i}Λ are 0 for i > 1.

Proof: This is immediate from Lemmas 12.8 and 5.7.

Corollary 12.10. The groups

Der^{i} = Der_{V}^{i} (O_{V ×_{M_{FGL,0}}U}, s_{∗}O_{V}) = Ext^{i}(Λ, s_{∗}O_{V})

vanish for i > 1. There is an exact sequence (which I think is canonically split, though the ext group of sheaves over M_{Ell,0} seems to be ℚ)

0 → O_{V} → Der^{0} → (Ω_{V}^{1} )^{−1} → 0

and an isomorphism

Der^{1} ≈ (Ω_{V}^{1} )^{−1}.

Proof: Since the homology groups of Λ are locally free sheaves, there are isomorphism

Der^{i} ≈ hom_{V ×_{M_{FGL,0}}U}(H_{i}Λ, s_{∗}O_{V})
≈ hom_{V} (s^{∗}H_{i}Λ, O_{V}).

Now there are canonical isomorphism

s^{∗}π_{2}^{∗}ω_{U} ≈ ω_{V}
s^{∗}π_{2}^{∗}Ω_{U}^{1} ≈ Ω_{V}^{1} .

The result then follows from Corollary 12.9.

12.3. Computations over ℤ. Now return to the situation of maps

U → M_{Ell} and V → M_{Ell} ∈ ob (M_{Ell})′_{et} ,

and a section s : V → V ×_{M_{Ell}} U. The results of the previous two

sections will now be assembled to give calculations of

H_{i}Λ = H_{i}Λ_{V ×_{M_{FGL}}U/V}

and

Der^{i} = Der_{V}^{i} (O_{V ×_{M_{FGL}}U}, s_{∗}O_{V})
= Ext^{i}(Λ, s_{∗}O_{V}).

Working remark 23. First some more notation.

Theorem 12.11. The sheaves H_{i}Λ are zero for i ≥ 2. There are exact sequences

0 → ℚ ⊗ hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) → H_{0}Λ → ℚ ⊗ π_{2}^{∗}Ω_{U}^{1} → 0

and

0 → ⨁_{p} ℚ/ℤ_{(p)} ⊗ Ω_{V}^{1} ⊗ v^{−1}Ô_{V} /v^{−1}O_{V}

→ H_{1}Λ → ℚ ⊗ π_{2}^{∗}Ω_{V}^{1} ⊗ hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) → 0.

Proof: Consider the long exact sequence coming from the cofibration

Λ → Λ ⊗ ℚ → Λ ⊗ ℚ/ℤ.

Theorem 12.12. The groups Der^{i} are zero for i = 0 and i > 2. There is an exact sequence

0 → M → Der^{1} → ∏_{p} (v^{−1}Ô_{V} /O_{V} )_{p} → 0

where M lies in the (splittable) exact sequence

0 → (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V} → M → ∏_{p}(O_{V})_{p}/O_{V} → 0.

ELLIPTIC COHOMOLOGY (February 22, 1999) 55

There is also an isomorphism

Der^{2} ≈ (Ω_{V}^{1} )^{−1} ⊗ ∏_{p}(O_{V})_{p}/O_{V} .

Remark 2. Note that Der^{2} is a vector space over ℚ.

Corollary 12.13. For each morphism

s : V → V ×_{M_{Ell}} U

of (M_{Ell})′_{et}, and each i ≥ 0, the map
Der_{V}^{i} (O_{V ×_{M_{FGL}}U}, s_{∗}O_{V}) → Der_{V}^{i} ((1 × π_{2})^{∗}O_{V ×_{M_{FGL}}U}, s_{∗}O_{V} )

≈ Der_{V}^{i} (O_{V ×_{M_{FGL}}V} , Δ_{∗}O_{V})

derived from

V ×_{M_{FGL}} U ⟶^{1×π_{2}s} V ×_{M_{FGL}} V
s↑  ↑^{Δ}
V  V

is an isomorphism.

13. Coherent cohomology of the stack M_{Ell}

Working remark 24. There is a lot more to get straight here. For the moment I am going to consider the stack which is covered by the union of the Hopf-algebroids

ℤ[a_{1}, …, a_{6}][c_{4}^{−1}] → ℤ[a_{1}, …, a_{6}][c_{4}^{−1}][r, s, t, λ^{±1}]
ℤ[a_{1}, …, a_{6}][Δ^{−1}] → ℤ[a_{1}, …, a_{6}][c_{4}^{−1}][r, s, t, λ^{±1}].

I will take this to be the definition of the stack M_{Ell}. This is the part of the moduli stack on which the function j is defined. It is also the part over which an affine etale open U over which ω is trivializable give rise to a Landweber exact cohomology.

13.1. When 6 is invertible. Let M be a coherent sheaf on M_{Ell} and suppose that 6 · 1_{M} is an isomorphism. Let

A = ℤ[\frac{1}{6}][γ_{4}, γ_{6}]
δ = \frac{1}{1728}(γ_{4}^{3} − γ_{6}^{2}) ∈ A

and write U_{0} = spec A[γ_{4}^{−1}], U_{∞} = spec A[δ^{−1}], and U_{0,∞} = U_{0} ∩ U_{∞} = spec A[(γ_{4}δ)^{−1}]. The curve

y^{2} = x^{3} + γ_{4}x + γ_{6}

becomes an elliptic curve when restricted to U_{0} and U_{∞}, and U_{0,∞} and so defines maps

U_{0} ⟶^{f_{0}} M_{Ell}
U_{∞} ⟶^{f_{∞}} M_{Ell}
U_{0,∞} ⟶^{f_{0,∞}} M_{Ell}.

It is not hard to check (details later) that

U_{0} ×_{M_{Ell}} U_{0} = spec O_{U_{0}}[λ^{±1}]
U_{∞} ×_{M_{Ell}} U_{∞} = spec O_{U_{∞}}[λ^{±1}]
U_{0} ×_{M_{Ell}} U_{∞} = spec A[(γ_{4}δ)^{−1}][λ^{±1}],

ELLIPTIC COHOMOLOGY (February 22, 1999) 57

and that each case, the the isomorphism π_{1}^{∗}A_{( −)} → π_{2}^{∗}A_{( −)} is given by

\begin{matrix}x & ↦ λ^{2}x \\ y & ↦ λ^{3}y \end{matrix} \begin{matrix}γ_{4} & ↦ η_{R}(γ_{4}) = λ^{4}γ_{4} \\ γ_{6} & ↦ η_{R}γ_{6} = λ^{6}γ_{6}. \end{matrix} (13.1)

For i = 0, i = ∞ or i = 0, ∞, let M_{i} be the module of global sectionf of f_{i}^{∗}M. Then each M_{i} comes equipped with an η_{R}-linear map

ψ : M_{i} → M_{i}[λ^{±1}].

Let N_{i} ⊂ M_{i} be the subset of elements m for which ψ(m) = m.

Proposition 13.2. If “multiplication by 6”: M → M is an isomorphism, then

H^{s}(M_{Ell}; M_{∗}) = 0 if s > 1.

There is an exact sequence

0 → H^{0}(M) → N_{0} ⊕ N_{∞} → N_{0,∞} → H^{1}(M) → 0.

For example, take M_{∗} to be the sheaf of graded rings

M_{∗} = ⨁_{n∈ℤ} ω^{n},

graded so that ω^{n} has degree 2n. The form dx/y defines over each U_{i} a nowhere vanishing section of u of ω. It follows from (13.1) that η_{R}(u) = λ^{−1}u. Let O_{∗} be the graded ring

O_{∗} = ℤ[\frac{1}{6}][c_{4}, c_{6}]
\begin{aligned}c_{4} & = u^{4}γ_{4} \\ c_{6} & = u^{6}γ_{6} \end{aligned}  \begin{aligned}|c_{4}| & = 8 \\ |c_{6}| & = 12. \end{aligned}

Then one easily calculates

N_{0} = O_{∗}[c_{4}^{−1}]
N_{∞} = O_{∗}[Δ^{−1}]
N_{0,∞} = O_{∗}[(c_{4}Δ)^{−1}].

It follows that H^{0}(M) = O_{∗}, and that H^{1} is concentrated in degrees ≤ 14.

Working remark 25. I think that

H^{0}(ω^{n}) ⊗ H^{1}(ω^{−7−n}) → H^{1}(ω^{−7}) → ℤ[\frac{1}{6}]

is an isomorphism. The last map is an isomorphism. The group is generated by c_{6}/(c_{4}Δ).

13.2. When 3 is nilpotent, and all geometric fibers are ordinary. Now suppose that 3 · 1_{M} = 0, and consider the elliptic curve

y^{2} = x^{3} + b′_{2}x^{2} + b′_{6}

over the ring

A = 𝔽_{3}[b′_{2}^{±1}, b′_{6}].

Set U = spec A. Then the curve above defines a map

f : U → M_{Ell}.

It is not hard to check that

U ×_{M_{Ell}} U = spec A[λ^{±1}].

As before, if we let M_{0} be the module of global sections of f^{∗}M, then M_{0} comes equipped with an η_{R}-linear map

ψ : M_{0} → M_{0}[λ^{±1}].

Let N ⊂ M_{0} be the subset of elements for which ψ(n) = n.

Proposition 13.3. With the above notation,if, 3 · 1_{M} = 0, and v_{1} : ω^{p−1} ⊗ M_{∗} → M_{∗} is an isomorphism, then

H^{s}(M_{Ell}; M_{∗}) = \begin{cases}N & s = 0 \\ 0 & s > 0. \end{cases}

Again consider the example of the sheaf of graded rings M = ⨁ ω^{n}. One easily calculates that H^{0}M is the ring

𝔽_{3}[b_{2}^{±1}, b_{6}]

where

b_{2} = u^{2}b′_{2} b_{6} = u^{6}b′_{6}.

ELLIPTIC COHOMOLOGY (February 22, 1999) 59

13.3. When 2 is nilpotent and all geometric fibers are ordinary. Suppose now that 2 · 1_{M_{∗}} is nilpotent, and that c_{4} : ω^{4} ⊗ M_{∗} → M_{∗} is an isomorphism. Let V = spec ℤ_{2}[α_{1}^{±1}, α_{6}] = spec A and let f : V → M_{Ell} be the map classifying the elliptic curve A with equation

y^{2} + α_{1}xy = x^{3} + α_{6}.

Define a G_{m}-action μ : G_{m} × V → V by

α_{1} ↦ λα_{1}
α_{6} ↦ λ^{6}α_{6}

Over the G_{m} × V, the map ϵ

\begin{aligned}ϵ(x) & = λ^{2}x \\ ϵ(y) & = λ^{3}y \end{aligned} (13.4)

sends a point on the elliptic curve π_{2}^{∗}A defined by the equation y^{2} + α_{1}xy = x^{3} + α_{6} to a point on the elliptic curve μ^{∗}A defined by the equation y^{2} + (λα_{1})xy = x^{3} + (λ^{6}α_{6}). This gives a 2morphism (also denoted ϵ) from f ∘ π_{2} to f ∘ μ, and therefore an isomorphism ϵ^{∗} : π_{2}^{∗}f^{∗}M → μ^{∗}f_{∗}M. The abelian group p_{∗}f^{∗}M therefore becomes a co-module over ℤ[λ^{±1}], and hence acquires a grading. The homogeneous part p_{∗}f^{∗}M[n] of degree n is the set of sections a of f^{∗}M satisfying μ^{∗}s = λ^{n}π_{2}^{∗}s. The image of H^{0}(M_{Ell}; M) in H^{0}(V; f^{∗}M) is contained in p_{∗}f^{∗}M[0].

Working remark 26. Name the structure maps. They are all called p right now. I guess I could avoid naming them altogether by using H^{0}M for p_{∗}M. Also, given f : X → Y and a sheaf M on Y, what is the notation for the map of sections of M to the sections of f^{−1}M and to the sections of f^{∗}M.

The line bundle f^{∗}ω has the nowhere vanishing section

u = \frac{dx}{2y + α_{1}x} = \frac{dy}{3x^{2} − α_{1}y} .

Under the map (13.4) it is sent to λ^{−1}u. It follows that “multiplication by u” gives an isomorphism

(p_{∗}f^{∗}M) [n] ≈ (p_{∗}f^{∗}ω ⊗ M) [n − 1].

The entire graded module p_{∗}f^{∗}M can therefore be identified with the λ = 1 eigenspace in ⨁ p_{∗}f^{∗}ω^{k} ⊗ M.

Next consider the scheme Z = spec A[s]/(s^{2} + α_{1}s), and define maps η_{R}, η_{L} : Z → V by

spec η_{L}(α_{1}) = α_{1}
spec η_{L}(α_{6}) = α_{6}

and

spec η_{R}(α_{1}) = α_{1} + 2s
spec η_{R}(α_{6}) = α_{6}.

Define a map ϵ : η_{L}^{∗}A → η_{R}^{∗}A by

x ↦ x
y ↦ y + sx

This gives a 2-morphism (also denoted ϵ) from f ∘ η_{L} → f ∘ η_{R}, and equips the sheaf f^{∗}M with a map ϵ^{∗} : η_{L}^{∗}f^{∗}M → η_{R}^{∗}f^{∗}M.

The ring O_{Z} is free of rank 2 as a module over O_{V}, with basis {1, s}. Let’s use the map “multiplication by s” to identify the cokernel of

η_{L}^{∗} : H^{0}(f^{∗}M) → H^{0}(η_{L}^{∗}f^{∗}M) = O_{Z} ⊗_{O_{V} ,η_{L}} H^{0}(f^{∗}M)

with H^{0}(f^{∗}M). Let d : H^{0}(f^{∗}M) → H^{0}(f^{∗}M) be the composite

H^{0}(f^{∗}M) ⟶^{η_{R}^{∗}} H^{0}(η_{R}^{∗}f^{∗}M) ⟶^{ϵ} H^{0}(η_{L}^{∗}f^{∗}M)

→ H^{0}(η_{L}^{∗}f^{∗}M)/H^{0}(f^{∗}M) ≈ H^{0}(f^{∗}M).

Lemma 13.5. The map d^{2} is 0.

Next we need to investigate the effect of the map d on the grading. Define an action of G_{m} on Z by

\begin{matrix}α_{i} & ↦ λ^{i}α_{i} \\ s & ↦ λs. \end{matrix}

With respect to this action, both maps η_{L} and η_{R} are equivariant.

Lemma 13.6. The map d sends an element of H^{0}(f^{∗}M)[n] to an element of H^{(}f^{∗}M)[n + 1].

Proof: The map d is essentially “multiplication by s”. The result follows easily. More details later.

Working remark 27. Now we have two choices. We can either modify the definition of d so that it is degree preserving, and a map from

H^{0}(f^{∗}ω^{n} ⊗ M)[0] → H^{0}(f^{∗}ω^{n+1} ⊗ M)[0]

ELLIPTIC COHOMOLOGY (February 22, 1999) 61

or just leave things as they are. I can’t decide which is better.

Proposition 13.7. Suppose that M is a quasi-coherent sheaf over M_{Ell} with 2^{k}1_{M} = 0 for some k, and with the property that c_{4} : ω^{4} ⊗ M → M is an isomorphism. Write

N_{k} = H^{0}(f^{∗}M)[k] ≈ H^{0}(f^{∗}ω^{k} ⊗ M)[0].

Thee cohomology group H^{t}(M) is naturally isomorphic to the t^{th} cohomology group of the complex

N_{0} ⟶^{d} N_{1} → ⋯ → N_{k−1} ⟶^{d} N_{k} → ….

Example 13.8. Consider the example of

M_{k} = c_{4}^{−1} ⨁_{n∈ℤ} ω^{n}O_{M_{Ell}}/2^{k}

In case k = 1 the differential d is zero since η_{L} = η_{R} mod 2. The ring H^{∗}M can be identified as the graded ring ℤ/2[a_{1}^{±1}, a_{6}][η], with a_{i} = u^{i}α_{i}, and η ∈ H^{1}(f^{∗}ω ⊗ M). The Bockstein operator is given by β(a_{1}) = η, and one can identify

H^{∗}(\underleftarrow{lim} M_{k})

with the subring of ℤ_{2}[a_{1}^{±1}, a_{6}, η]/(2η) generated by {a_{1}^{2}, a_{6}, η, a_{1}η}. As an abstract ring it is isomorphic to

ℤ_{2}[a_{1}^{±2}, a_{6}, η, β]/(2η, β^{2} − a_{1}^{2}η^{2}).

Example 13.9. Another important example is the case in which

M_{k} = c_{4}^{−1} ⨁_{n∈ℤ} ω^{n}ˆ(O_{M_{Ell}}/2^{k})

where the “hat” indicates completion along the supersingular locus. The ring of global sections of f^{∗}M is

a_{1}^{−1}ℤ/2^{k}[[a_{1}]][a_{6}, u^{±1}],

where u ∈ H^{0}(f^{∗}ω) is the invariant section mentioned above. It is not difficult to calculate that

H^{∗}(f^{∗} \underleftarrow{lim}_{k}M_{k})

is isomorphic to the subring of a_{1}^{−1}ℤ_{2}[[a_{1}]][a_{6}, η]/(2η) which is topologically generated by {a_{1}^{2}, a_{6}, η, a_{1}η}. As an abstract ring it is

a_{1}^{−2}ℤ_{2}[[a_{1}^{2}]][a_{6}, η, β]/(2η, β^{2} − a_{1}^{2}η^{2}).

Working remark 28. The next example better include Ω^{1}. Then you better mention the variety “cusps”. I this case it is defined by α_{6} = 0.

Example 13.10. Putting these together, we find that we can calculate

ELLIPTIC COHOMOLOGY (February 22, 1999) 63

Appendix A. Low dimensional cohomology of categories

Consider the following situation. Suppose that C and D are categories with the same object set, and that F : C → D is a functor. Given a morphism f ∈ D(d_{1}, d_{0}) let

C(d_{1}, d_{0})_{f}

be the subset of g ∈ C(d_{1}, d_{0}) with F g = f. In addition assume that the functor F creates isomorphisms (ie. a map g is an isomorphism if and only if F G is). Then the set C(d_{0}, d_{0})_{1} is a group, and it acts on the left of C(d_{1}, d_{0}). Suppose that for each pair of objects d_{0} and d_{1}, the set C(d_{1}, d_{0})_{f} is a principal homogeneous space for the group C(d_{0}, d_{0})_{1}. It will also make life easier if we assume that this group is abelian.

We wish to investigate the obstruction to the existence of a section, the enumeration of sections, and the automorphism group of a section. It is useful consider the case when the categories C and D have only one object, and the morphism sets form a group. In this case the situation is that of a surjective map of groups with abelian kernel. The obstruction to the existence of a section is H^{2}, the set of sections up to vertical equivalence is a principal homogeneous space for H^{1}, and the automorphism group of any section is H^{0}.

Working remark 29. Many thanks to Charles Rezk here.

Let’s write A(d) for the (abelian) group C(d, d)_{1}. The first thing to notice is that d ↦ A(d) is a covariant functor on D in a natural way. Indeed, suppose that f : d_{1} → d_{0} is a map in C and pick f_{1} ∈ C(d_{1}, d_{0})_{f}. Then by assumption, given α ∈ A(d_{1}) there is a unique β ∈ A(d_{0}) for which the following diagram commutes:

d_{1} ⟶^{f_{1}} d_{0}
α↓  ↓^{β}
d_{1} ⟶_{f_{1}} d_{0}.

Define

A(f) : A(d_{1}) → A(d_{0})

by A(f)(α) = β.

It is easily checked that A(f) is a homomorphism and it remains to verify its independence of the choice of f_{1}. For this, suppose that f_{2} is an other element of C(d_{1}, d_{0})_{f}. Then f_{2} = γf_{1} for some γ ∈ A(d_{0}). Let’s temporarily write A_{i}, i = 1, 2 for the map based on the choice f_{i}. Then A_{i}(α) is the unique solution to the equation

A_{i}(α)f_{i} = f_{i}α.

Now compute

A_{2}(α)f_{2} = f_{2}α
A_{2}(α)γf_{1} = γf_{1}α
γ^{−1}A_{2}(α)γf_{1} = f_{1}α
A_{2}(α)f_{1} = f_{1}α,

where the last step makes use of the commutativity of the A(d). It follows that A_{2}(α) satisfies the defining equation of A_{1}(α) and so the two must coincide.

Suppose instead that we are given a functor A from D to the category of abelian groups. Form the category

A ⋊ D

whose objects are the objects of D and whose morphisms from d_{1} to d_{0} are the set A(d_{0}) × D(d_{1}, d_{0}). The composition law is:

(β, g) ∘ (α, f) = (β · A(g)(α), g ∘ f).

The construction of this category does not make use of the fact that the groups A(d) are commutative.

The category A ⋊ D is an abelian group object in the category of categories over D with a fixed object set. The category C is a principal homogeneous space for A ⋊ D and so it follows that the obstruction to the existence of a section is in Quillen H^{1} of D with values in the abelian group object A ⋊ D. If this obstruction vanishes, then the set of sections is, by defintion, Quillen H^{0}.

In this situation, there is, for i > 0, an isomorphism of Quillen H^{i} with \underleftarrow{lim}^{i+1}A. The isomorphism is easily established by imitating the discussion of Schur in the case of groups.

Suppose then that we wish to write down a section of F. Choose then, for each f ∈ D(d_{1}, d_{0}) a map s(f) ∈ C(d_{1}, d_{0})_{f}. Given a

ELLIPTIC COHOMOLOGY (February 22, 1999) 65

composable pair

d_{0} ⟵^{f_{0}} d_{1} ⟵^{f_{1}} d_{2},

let δ(f_{0}, f_{1}) ∈ A(d_{0}) be the unique solution to the equation

d_{2} ⟶^{s(f_{1})} d_{1}
s(f_{2}f_{1})↓  ↓^{s(f_{0})}
d_{0} ⟶^{δ(f_{0},f_{1})} d_{0}.
δ(f_{0}, f_{1})s(f_{0}f_{1}) = s(f_{0})s(f_{1}).

By considering the situation

d_{0} ⟵^{f_{0}} d_{1} ⟵^{f_{1}} d_{2} ⟵^{f_{2}} d_{3}

it is easy to check that the function δ satisfies the “cocycle” condition

Af_{0}(δ(f_{1}, f_{2})) + δ(f_{0}, f_{1}f_{2}) = δ(f_{0}, f_{1}) + δ(f_{0}f_{1}, f_{2}), (A.1)

where the symbol “+” has been used to indicate composition, in order to emphasize that the group A(d_{0}) is abelian. Conversely, given a “cocycle” δ it is easy to construct a category C as above. Now let’s compare two choices s_{1} and s_{2} of s. For each f ∈ D(d_{1}, d_{0}) there is a unique α(f) ∈ A(d_{0}), with s_{2}(f) = α(f)s_{1}(f). Let δ_{i}, i = 1, 2 be the associated cocycles. Then

δ_{2}(f_{0}, f_{1})s_{2}(f_{0}f_{1}) = s_{2}(f_{0})s_{2}(f_{1})
δ_{2}(f_{0}, f_{1})α(f_{0}f_{1})s_{1}(f_{0}f_{1}) = α(f_{0})s_{1}(f_{0})α(f_{1})s_{1}(f_{1})
δ_{2}(f_{0}, f_{1})α(f_{0}f_{1})s_{1}(f_{0}f_{1}) = α(f_{0})Af_{0}(α(f_{1})) ∘ s_{1}(f_{0})s_{1}(f_{1})

and so

δ_{2}(f_{0}, f_{1})α(f_{0}f_{1}) = α(f_{0})Af_{0}(α(f_{1})) ∘ δ_{1}(f_{0}, f_{1}).

Define an equivalence relation on the set of cocycles δ by declaring δ_{1} to be equivalent to δ_{2} if there is a function α with

δ_{2}(f_{0}, f_{1}) + α(f_{0}f_{1}) = α(f_{0}) + Af_{0}(α(f_{1})) + δ_{1}(f_{0}, f_{1}). (A.2)

Then it is not hard to check that the set of equivalence classes of “extensions” of D by A is naturally in one to one correspondence with the set of cocycles δ modulo the equivalence relation (A.2).

Now suppose that functor C → D admits a section. Given a section s one easily constructs an equivalence of categories

C → A ⋊ D

by sending f̃ ∈ C(d_{1}, d_{0})_{f} to (α, f), where α is the unique solution to the equation

αs(f) = f̃.

Suppose now that s_{1} and s_{2} are two sections of F. Write

s_{1}(f) = α(f)s_{2}(f).

It follows from the fact that s_{1} and s_{2} preserve the composition law that α satisfies the “crossed homomorphism” identity

α(f_{0}f_{1}) = Af_{0}(α(f_{1})) + α(f_{0}). (A.3)

The set of sections is thus a principal homogeneous space for the abelian group of functions α satisfying the identity (A.3).

Next consider the situation where sections s_{1} and s_{2} are related by a vertical natural equivalence. Then there is a function d → β(d) ∈ A(d) representing the natural transformation.

Given f : d_{1} → d_{0}, the natural transformation property

s_{1}(d_{1}) ⟶^{β(d_{1})} s_{2}(d_{1})
s_{1}(f)↓  ↓^{s_{2}(f)}
s_{1}(d_{0}) ⟶_{β(d_{0})} s_{2}(d_{0})

becomes the identity

\begin{aligned}β(d_{0})s_{1}(f) & = s_{2}(f)β(d_{1}) \\ & = Af (β(d_{1})) s_{2}(f). \end{aligned} (A.4)

It follows that the function α(f) satisfies the equation

Af(β(d_{1})) + α(f) = β(d_{0}). (A.5)

The set of sections of F up to vertical natural equivalence can therefore be identified with the quotient of the abelian group of functions α satisfying (A.3), modulo the subgroup consisting of those satisfying the identity (A.5) for some β.

ELLIPTIC COHOMOLOGY (February 22, 1999) 67

Finally, consider the above situation with s_{1} = s_{2} = s. Then α(f) = 0, and equation (A.5) becomes

Af(β(d_{1})) = β(d_{0}).

In other words β is an element of \underleftarrow{lim}^{0}A. It follows that the group of “vertical” automorphisms of any section is naturally isomorphic to \underleftarrow{lim}^{0}A.

The situation becomes much clearer when expressed in terms of the cosimplicial replacement [2, X1,§5] of the functor A. The cosimplicial replacement C^{•}(D; A) is a cosimplical abelian group whose s^{th} cohomology group (or, more precisely, cohomotopy group) is the derived functor \underleftarrow{lim}^{s}A. It is defined by

C^{n} = C^{n}(D; A) = ∏_{\underline{f}} A(d_{0}) n > 0

where

\underline{f} = d_{0} ⟵^{f_{0}} d_{1} ⟵^{f_{1}} ⋯ d_{n−1} ⟵^{f_{n−1}} d_{n}

is a sequence of composable maps. It is best to think of elements of C^{n} as “functions” α on the set of composable n-tuples of maps of D (when n = 0 this is interpreted as “functions” on the set of objects). The coface maps are given for α ∈ C^{n} with n > 0 by

d^{i}α(f_{0}, …, f_{n}) = \begin{cases}Af_{0} (α(f_{1}, …, f_{n})) & i = 0 \\ α(…, f_{i−1}f_{i}, …) & 0 < i ≤ n \\ α(f_{0}, …, f_{n−1}) & i = n + 1, \end{cases}

and for α ∈ C^{0} by

d^{0}α(f_{0}) = Af_{0} (α(d_{1}))
d^{1}α(f_{0}) = α(d_{0}).

The codegeneracy maps are give on α ∈ C^{n+1} by

s^{i}α(f_{0}, …, f_{n}) = α(…, 1_{d_{i}}, f_{i}, …).

Now it is easy to check that the cocycle condition (A.1) on a function δ is precisely the condition that δ be a 2-cocycle in NC^{2}. The equivalence relation (A.2) is the condition that δ_{1} and δ_{2} differ by a coboundary. Thus the set of “extensions” of D by A can be identified with \underleftarrow{lim}^{2}A.

Given a section s : D → C it is not hard to check that any other section differs by a 1-cocycle in NC^{1}. It follows that the set of sections is precisely the set of 1-cocycles. Two sections are vertically equivalent if and only if the corresponding 1-cocycles differ by a coboundary. It follows that the set of equivalence classes of sections can be identified with \underleftarrow{lim}^{1}A.

This discussion is summarized in the following proposition

Proposition A.6. With the notation of this appendix, the functor

F : C → D

is classified by an element of \underleftarrow{lim}^{2}A. The vanishing of this class is a necessary and sufficient condition for the existence of a section. The set of sections is a principal homogeneous space for the group of 1-cocycles in the cosimplicial replacement of the functor A. The set of sections modulo (vertical) natural equivalence is a principal homogeneous space for \underleftarrow{lim}^{1}A, and the group of vertical automorphisms of any section is canonically isomorphic to \underleftarrow{lim} A.

ELLIPTIC COHOMOLOGY (February 22, 1999) 69

Appendix B. A spectral sequence for A_{∞}-maps

B.1. The spectral sequence. Let’s start with a ring spectrum F satisfying the following condition of Adams [1, Condition 13.3 (page 284)]

(1) Each map of spectra

X → F

with X finite, factors as

X → V → F

where V is finite and F_{∗}DV ≈ F^{∗}V is projective over F_{∗}.

We are now going to describe a simplicial model category structure on the category sA_{∞} of simplicial A_{∞}-rings, that will make systematic the kind of resolutions we wish to use. This construction is somewhat ad hoc. It is chosen so that the following result holds.

Proposition B.1. The F^{α} model category structure on sA_{∞} satisfies

i) If X_{•} → Y_{•} is a weak equivalence (resp. fibration), then for each t the maps

F_{t}X_{•} → F_{t}Y_{•}
π_{t}X_{•} → π_{t}Y_{•}

are weak equivalences (resp. fibrations) of simplicial groups;

ii) If W_{•} is cofibrant, then each W^{s} is a free A_{∞} ring on a spectrum V^{s} which has the property that F_{∗}V^{s} is a projective F_{∗}- module.

Remark 3. It follows from the second property that F_{∗}W^{s} is the free associative F_{∗}-algebra on F_{∗}V^{s} and that the maps

F^{∗}W^{s} → hom_{F_{∗}}(F_{∗}W^{s}, F_{∗})
π_{t}A_{∞}(W^{s}, F) → hom_{F_{∗}}(F_{∗}V^{s}, F̃^{0}S^{t})

are isomorphisms.

This result will be proved in B.4. Given this result, we are in position to investigate the function space A_{∞}(E, F) when the ring spectrum underlying F satisfies the condition above. To do this, regard E and S^{0} as constant simplicial objects in sA_{∞} and factor the unit map

S^{0} → W_{•} → E

as a cofibration followed by an acyclic fibration. By the homotopy spectral sequence (which collapses) the map

|W_{•}| → E

is a weak equivalence (of cofibrant-fibrant objects), so the map A_{∞}(E, F) → A_{∞}(|W_{•}|, F) is a weak equivalence, and we can get a spectral sequence by writing the latter as

Tot(A_{∞}(W_{•}, F)).

For simplicity of notation, lets write M^{•} for the cosimplicial space A_{∞}(W_{•}, F).

To define the E_{2}-term of the spectral sequence, we need to pick an element of π_{0}M^{0} which is equalized under the maps π_{0}d^{0} and π_{0}d^{1}. Now by our assumptions, the set π_{0}M^{k} is naturally isomorphic to the set of F_{∗}-algebra maps from F_{∗}W_{k} to F_{∗}. It follows that the equalizer of π_{0}d^{0} and π_{0}d^{1} is the set of F_{∗}-algebra maps from the coequalizer in F_{∗} algebras of

F_{∗}W_{1} ⇉^{F_{∗}d_{0}}_{F_{∗}d_{1}} F_{∗}W_{2}.

Since this comes from a simplicial diagram this is a reflexive coequalizer and so the F_{∗}-module which underlies the algebra coequalizer coincides with the coequalizer of the underlying diagram of F_{∗}-modules. By part i) of Proposition B.1 this is simply F_{∗}E. So in order to define the E_{2}-term of the spectral sequence, we have to pick an F_{∗}-algebra map from F_{∗}E to F_{∗}.

Having done this we now have a map of simplicial F_{∗}-algebras from F_{∗}W_{•} to F_{∗}, and we also get a “basepoint” in each M^{k}. With this choice of basepoint, there is a natural identification

π_{t}M^{k} ≈ Der_{F_{∗}}(F_{∗}W_{k}, F̃^{0}S^{t}).

The E_{2}^{s,t}-term of the spectral sequence is π^{s}π_{t}M^{•}. Now, by assumption, the map

F_{∗}W_{•} → F_{∗}E

is a weak equivalence of F_{∗}-algebras. Furthermore, each F_{∗}W_{k} is a free associative F_{∗}-algebra on a projective F_{∗}-module. It follows

ELLIPTIC COHOMOLOGY (February 22, 1999) 71

from the definition of Quillen derived functors, that the E_{2}-term of the spectral sequence is

Der_{F_{∗}}^{s} (F_{∗}E, F̃^{0}(S^{t})) .

B.2. The Reedy model category structure. Let Δ be the usual category of finite ordered sets, and let Δ_{≤n} be the full subcategory with objects

[0], [1], …, [n].

If C is a category with finite limits and colimits, then the restriction functor

res_{n}^{op} : C^{Δ^{op}} → C^{Δ_{≤n}^{op}}

has both a left and a right adjoint. Let’s denote the left adjoint l_{n}^{op} and write skel_{n} for the composite l_{n}^{op}res_{n}^{op}. The functor skel_{n} is the n-skeleton functor.

Let sC be the category C^{Δ^{op}} of simplicial objects in C.

Definition B.2. Let X_{•} be an object of sC. The n^{th} latching object of X_{•} is object (skel_{n−1} X_{•})_{n}.

Now let cC be the category of cosimplicial objects in C. As above, the restriction functor

res_{n} : C^{Δ} → C^{Δ_{≤n}}

has both a left and a right adjoint. Let’s call the right adjoint r_{n}, and write coskel^{n} for the composite r_{n}res_{n}. The functor coskel^{n} is the n^{th} coskeleton functor.

Definition B.3. Let X^{•} be an object of cC. The n^{th} matching object of X^{•} is object (coskel^{n−1} X^{•})_{n}.

We can now define the Reedy model category structure. …

B.3. The P model category structure. This is the variation of the E_{2} model category structure described in [5, §5.9]. Suppose that C is a closed (pointed) model category, with arbitrary colimits in which every object is fibrant. Suppose also that M ∈ ob C is a cofibrant co-grouplike object of C.

Definition B.4. [5, 5.4–5.6] A map g_{•} : X_{•} → Y_{•} of sC is a (1) weak M-equivalence if for each j ≥ 0 the map

ho C(Σ^{j}M, X_{•}) → ho C(Σ^{j}M, Y_{•})

is a weak equivalence of simplicial groups

(2) M-fibration if it is a Reedy fibration and if for each j ≥ 0 the map

ho C(Σ^{j}M, X_{•}) → ho C(Σ^{j}M, Y_{•})

is a fibration of simplicial groups;

(3) M-cofibration if it is a retract of an “M-free” map, where a map X_{•} → Y_{•} M-free if for each n ≥ 0 there is a cofibrant object Z_{n} ∈ ob C which is weakly equivalent to a coproduct of copies of Σ^{j}M (j ≥ 0), and a map Z_{n} → Y_{n} such that the induced map

(X_{n} ∐_{L_{n}X_{•}} L_{n}Y_{•}) ∐ Z_{n} → Y_{n}

is a trivial cofibration.

Proposition B.5. [5, 5.3] The category sC equipped with M weak equivalences, M fibrations, and M cofibrations, and the natural simplicial structure, is a closed simplicial model category.

Remark 4. (1) It isn’t really necessary that C be pointed. The object M, being co-goup like, comes equipped in ho C with a “cobasepoint” map to the initial object, representing the identity element of the corresponding group of maps. One needs to redefine ΣM as the pushout of CM ← M → CM, where CM is an object obtained by factoring a representative of this “co-basepoint” into a cofibration followed by an acyclic fibration.

(2) If one has in mind a set P of cofibrant co-grouplike objects, then one can define a P -model category structure by requiring the condition on M to hold for each element of P. Of course this is no gain in generality, since the P -model category structure coincides with the M-model category structure with M taken to be the coproduct of the objects of P.

(3) The way it will come up in the present paper, the set P will be stable under the suspension operator. This means that the clauses “for each j ≥ 0” reduce to the case j = 0 and needn’t be considered.

ELLIPTIC COHOMOLOGY (February 22, 1999) 73

(4) If M and M′ are weakly equivalent then the M and M′ model category structures coincide.

(5) It follows that the set P doesn’t quite need to be a set. It is only necessary that the collection of weak equivalence classes of elements of P be small, since the P model category structure can be built using one cofibrant representative from each weak equivalence class.

B.4. The F^{α} model category structure. Let T be an operad over the linear isometries operad, and T-alg the corresponding category of algebras. The category T-alg is a closed topological model category and is generated by small objects. It has all (small) enriched limits and colimits, and has the property that every object is fibrant.

For a ring spectrum F, let F^{α} be the collection consisting of spectra V which are weakly equivalent to a finite spectrum and for which F_{∗}V is a projective module over F_{∗}.

Definition B.6. The F^{α} model category structure on s T-alg is the P model category structure, where

P = {T (V) | V ∈ F^{α}} .

The spectra of the form T (V) are co-grouplike since

ho T-alg(T (V), Y) = ho S(V, Y)

is an abelian group.

If F satisfies the condition of Adams then [1, Lemma 13.8 (page 287)]) shows that given any spectrum X and a homology class in x ∈ F_{∗}X there is a map U → X, U ∈ F^{α}, and an element e ∈ F_{∗}U with the property that x is in the image of

ho S(U, X) → F_{∗}X
f ↦ F_{∗}f(e).

More generally suppose that X : I → S is a diagram in indexed by a category I which has only finitely many objects and morphisms. Define F_{∗}X to be the diagram of F_{∗}-modules

i ↦ F_{∗}Xi.

Then given any diagram I-diagram V of finite sets, and a map x : V → F_{∗}X there is an element U ∈ F^{α}, an element e ∈ F_{∗}U with the property that x factors through the resulting map

ho S(U, X) → F_{∗}X
ho S(U, X) = i ↦ ho S(U, Xi).

The proof follows the argument of [1, Lemma 13.8] and makes use of the fact that, since I is finite and V takes its values in finite sets, the functor E ↦ S^{I}(V, E_{∗}X) commutes with directed homotopy colimits.

In the situation that arises below, X is a simplicial spectrum and V is a simplicial set with only finitely many non-degenerate simplices. Though in this case the indexing category is not finite, the fact that V has only finitely many non-degenerate simplices means that it is of the form l_{n}^{op}V′ for some n, and so the diagram hom set can be calculated as a hom set in S^{Δ_{≤n}^{op}}.

Proof of Proposition B.1: Part ii) is easy and follows from the Kunneth spectral sequence (which collapses, since projective modules are flat)

In part i), by factoring the map X_{•} → Y_{•} into an acyclic Reedy cofibration followed by a Reedy fibration, we may in both cases assume that the map is an F^{α} fibration. In both cases we need to produce a lift in a diagram of the form

V ⟶ F_{t}X_{•}
↓  ↓
D ⟶ F_{t}Y_{•}.

In the “fibration” case, the map V → D runs through the inclusions V [n, k] ⊂ Δ[n] while in the “weak equivalence” case it runs through the inclusions Δ̇[n] ⊂ Δ[n]. Since V and D have only finitely many non-degenerate simplices, there is a U ∈ F^{α}, and an element of F_{∗}U with the property that the diagram factors as

V ⟶ ho A_{∞}(T U, X_{•}) ⟶ F_{t}X_{•}
↓  ↓  ↓
D ⟶ ho A_{∞}(T U, Y_{•}) ⟶ F_{t}Y_{•},

where we have made use of the isomporphism

ho A_{∞}(T U, E) ≈ ho S(U, E).

ELLIPTIC COHOMOLOGY (February 22, 1999) 75

By the definition of the F^{α} model category structure there is a lift in the left square. Composing this with the rightmost top arrow gives the desired lift.

Appendix C. A calculation: so I don’t have to keep reconstructing this

Notation:

f : E → F a map of elliptic spectra spec π_{0}E = U → M_{Ell}

spec π_{0}F = V → M_{Ell}

spec π_{0}f = s : V → V ×_{M_{FGL}} U

π : V^{ord} ×_{M_{FGL}} U^{ord} → V^{ord} × U^{ord}

The map s is given to us factored through V ×_{M_{Ell}} U.

Now we calculate, twice, in two notations, all reduced mod p^{m}, but not indicated:

Rhom_{π_{0}F ∧E} (Λ_{π_{0}F ∧E/π_{0}F} , π_{0}F )

= Rhom_{π_{0}F ∧E} (Λ_{π_{0}v_{1}^{−1}(F ∧E)/π_{0}v_{1}^{−1}F} , π_{0}F )

(M = Σ^{−1}π_{0}L_{K(2)}F/F)

= Rhom_{π_{0}v_{1}^{−1}F ∧E} (Λ_{π_{0}v_{1}^{−1}(F ∧E)/π_{0}v_{1}^{−1}F} , M)

= Rhom_{π_{0}v_{1}^{−1}F ∧E} (Λ_{π_{0}v_{1}^{−1}(F ∧E)/π_{0}v_{1}^{−1}F} , M)

= Rhom_{π_{0}v_{1}^{−1}F ⊗π_{0}v_{1}^{−1}E} (Λ_{π_{0}v_{1}^{−1}F ⊗π_{0}v_{1}^{−1}E/π_{0}v_{1}^{−1}F} , M)

= Rhom_{π_{0}v_{1}^{−1}F ⊗π_{0}v_{1}^{−1}E} (π_{0}v_{1}^{−1}F ⊗ Ω^{1} (π_{0}v_{1}^{−1}E) , M)

= Rhom_{π_{0}v_{1}^{−1}E} (Ω^{1} (π_{0}v_{1}^{−1}E) , M)

= Rhom_{π_{0}v_{1}^{−1}F} (Ω^{1} (π_{0}v_{1}^{−1}F ) , M)

= (Ω^{1})^{−1} ⊗_{π_{0}F} M = (ω^{2}(cusps))^{−1} ⊗_{π_{0}F} M

= ω^{−2}(−cusps) ⊗ M

Here’s another good one

ELLIPTIC COHOMOLOGY (February 22, 1999) 77

Lemma C.1. Suppose that E is the p-adic completion of an etale elliptic spectrum. There is, after p-adic completion, a fibration

∏_{cusps} Ω_{0}^{∞}Σ^{4}KU → A_{∞} {E} → Ω_{0}^{∞}Σ^{4}L_{K(2)}E/E

Where Ω_{0}^{∞} means the connected component. In particular, the space A_{∞} {E} is connected.

Working remark 30. I am guessing here. I guess I know this fibration up to homotopy.

Proof: The proof makes use of the following isomorphisms

ΩA_{∞} {E}_{s} ≈ A_{∞}(E, E)
π_{t−1}A_{∞}(E, E) ≈ π_{t}L_{K(2)}E/E ⊗ (Ω_{U}^{1})^{−1}

(the spectral sequence)

≈ π_{t}L_{K(2)}E/E ⊗ ω(cusps)^{−2}

(Kodaira-Spencer)

≈ π_{t−4}L_{K(2)}E/E NOT

Appendix D. Temporary ramblings to work something out

I will not distinguish in notation between a sheaf over an affine scheme and its global sections.

I am currently stumped with this calculation of Λ given in section 12. Thus let U and V be objects of (M_{Ell})′_{et}, and set

Λ = Λ_{V ×_{M_{FGL}}U/V} .

The following has been established:

H_{0}Λ ⊗ ℤ/p^{k} ≈ i_{∗}π_{2}^{∗}Ω_{(U⊗ℤ/p^{k})^{ord}}^{1} (D.1)

H_{i}Λ ⊗ ℤ/p^{k} = 0 i > 0 (D.2)

hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) ⊗ ℚ ↣ H_{0}Λ ⊗ ℚ ↠ π_{2}^{∗}Ω_{U}^{1} ⊗ ℚ (D.3)

H_{1}Λ ⊗ ℚ = hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) ⊗ π_{2}^{∗}Ω_{U}^{1} ⊗ ℚ (D.4)

H_{i}Λ ⊗ ℚ = 0 i > 1. (D.5)

We also know, by definition, that H_{i}Λ = 0 for i < 0. It ought to be easy to work out the sheaves H_{i}Λ from this. Let’s consider the exact sequence which results from the triangle

(cofibration)

Λ → Λ ⊗ ℚ → Λ ⊗ ℚ/ℤ.

It immediately gives

H_{i}Λ = 0 i > 1 (D.6)

H_{1}Λ ≈ hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) ⊗ π_{2}^{∗}Ω_{U}^{1} ⊗ ℚ (D.7)

H_{0}Λ ↣ H_{0}Λ ⊗ ℚ ↠ H_{0}Λ ⊗ ℚ/ℤ. (D.8)

The point at which I am confused is with this last exact sequence. I can’t seem to get the last map to be surjective.

So, let’s see how to compute this map. Let’s try and go from

hom(π_{1}^{∗}ω_{V} , π_{2}^{∗}ω_{U}) ⊗ ℚ → \underrightarrow{lim} i_{∗}π_{2}^{∗}Ω_{(U⊗ℤ/p^{k})^{ord}}^{1}.

There is a natural isomorphism H_{0}Λ_{X/S} ≈ Ω_{X/S}^{1} for flat X/S.

ELLIPTIC COHOMOLOGY (February 22, 1999) 79

Appendix E. Calculation

I am kind of stuck again, and so here is another appendix. First some notation. For this section I am working mod a power of a prime p.

• (M_{Ell})′_{et} is the category I have been using. It consists of etale opens over the elliptic moduli stack which are affine, and over which the line bundle ω is trivializable.

• M_{Ell,p}^{ss} ⊂ M_{Ell} is the closed substack of consisting of curves, all of whose geometric fibers are supersingular.

• The stack M_{Ell,p}^{ord} is the complement of M_{Ell,p}^{ss}. It consists of curves, all of whose geometric fibers are ordinary.

• The categories (M_{Ell,p}^{ss})′_{et} and (M_{Ell,p}^{ord})′_{et} are the full subcat-

egories of (M_{Ell})′_{et} consisting of elliptic curves, all of whose geometric fibers are supersingular (resp. ordinary).

Remark 5. There are three different ext groups that will arise below. Suppose that U = spec R is an affine scheme, and that M and N are quasi-coherent sheaves of R-modules. Then

• Ext_{R}^{i}(M, N) = Ext_{Γ(O_{U})}(Γ_{U}M, Γ_{U}N) denotes “ext” in the category of R-modules;

• Ext_{U}^{i}(M, N) denotes “ext” in the abelian category of sheaves of modules over U;

• \underline{Ext}_{U}^{i}(M, N) is the i^{th} derived functor of sheaf hom. Is it the sheaf associated to the presheaf Ext_{V}^{i} (M|_{V} , N|_{V})?

Working remark 31. The substack M_{Ell,p}^{ss} has, locally to be given a subscheme structure. I don’t want to take it’s reduced induced subscheme structure, since I want it to be flat over the ground ring. I guess I need to “define” it by regarding E_{p−1} as a map from ω^{1−p} to ω. I don’t quite know what this means when p = 2. I guess it means that the “supersingular locus” doesn’t lift to a flat closed substack at 2 or 3. Hmm, actually, the structure sheaf of this substack does not seem to play a role in what follows, so I can take the reduced induced subscheme structure.

Now the substack M_{Ell,p}^{ss} is locally defined by the vanishing of a single function. The associated sheaf of ideals is a line bundle over M_{Ell}, and can naturally be identified with ω^{1−p}. That is to say, there is, (modulo p), an exact sequence ω^{1−p} ⟶^{v_{1}} O → O_{M_{Ell,p}^{ss}}.

Locally we can find a nowhere vanishing section s of ω^{1−p}. The element v_{1}(s) is then a function defining the supersingular locus. Modulo a power of p the above still holds, provided ≠ 2, 3.

We are interested in computing the derived functors of the inverse limit over (M_{Ell})′_{et} of

π_{2t+1} Bhaut_{1} E_{U} = hom_{O_{U}} (O_{U^{ord}}, ω^{t} ⊗ (Ω^{1})^{−1})
π_{2t} Bhaut_{1} E_{U} = Ext_{Γ(O_{U})}(O_{U^{ord}}, ω^{t} ⊗ (Ω^{1})^{−1})

Now the first these contravariant functors is the restriction of (M_{Ell})′_{et} sheaf i_{!}O_{M_{Ell,p}^{ord}}. The second is not the restriction of a sheaf, and must be approached differently.

Thus let’s restrict our attention to an open U, and let i : U^{ss} → U and j : U^{ord} → U be the inclusion of the supersingular and ordinary loci. Let’s write F for the “hom” sheaf, and G for the “ext” sheaf. Let’s also choose a function s which defines the supersingular locus. Then F and G can be identified with the sheaves

\underleftarrow{lim} s^{n}ω^{t} ⊗ (Ω^{1})^{−1}  and
\underleftarrow{lim}^{1}s^{n}ω^{t} ⊗ (Ω^{1})^{−1}.

There are two spectral sequences for computing the group of sheaf extensions Ext^{n}(O_{U^{ord}}, ω^{t} ⊗ (Ω^{1})^{−1}). One reduces to the short exact sequence

\underleftarrow{lim}^{1}s^{k} · H^{n−1}(ω^{t} ⊗ (Ω^{1})^{−1}) → Ext^{n} → \underleftarrow{lim} s^{k} · H^{n}(ω^{t} ⊗ (Ω^{1})^{−1})

and the other reduces to the long exact sequence

⋯ → H^{n−2} \underleftarrow{lim}^{1} → H^{n} \underleftarrow{lim} → Ext^{n} → H^{n−1} \underleftarrow{lim}^{1} → ….

Since U is affine, the groups H^{n} are zero for n > 0 and so the above data reduces to

Ext^{0} ≈ \underleftarrow{lim}^{0}H^{0} ≈ H^{0} \underleftarrow{lim}^{0}

\underleftarrow{lim}^{1}H^{0} ≈ Ext^{1}

H^{1} \underleftarrow{lim}^{0} ↣ Ext^{1} ↠ H^{0} \underleftarrow{lim}^{1}

H^{i} \underleftarrow{lim} = 0 i > 1

H^{i} \underleftarrow{lim}^{1} = 0 i > 0.

(The last two isomorphisms follow from the Milnor sequences).

ELLIPTIC COHOMOLOGY (February 22, 1999) 81

We can compute H^{0} \underleftarrow{lim}^{0} from the exact sequence of a “pair.” We can compute \underleftarrow{lim}^{1}H^{0} as well, so we can get H^{0} lim^{1}. The inverse limit coincides with the sheaf cohomology for the lim 1 sheaf, so we get the derived functors of inverse limit in this case. For the lim sheaves use the spectral sequence relating sheaf to Čech cohomology, which in this case reduces to a long exact sequence.

Appendix F. Equivalence of realization spaces

I think I can simplify some of the homological algebra by passing to a refined covering. This is an exploration of that possibility.

First some notation. What I have been calling (M_{Ell})′_{et} I think I will call Et′ /M_{Ell}, and for the purposes of this appendix I will abbreviate it with an I. We have constructed a diagram D : I → ho A_{∞}.

Lemma F.1. Suppose that f : J → I is the inclusion of a full subcategory, with the properties that (1) each U ∈ ob I is covered by an object of J, and (2) if V ∈ ob J and U ⊂ V, then U ∈ ob J. If X is a fibrant realization of D, then the natural map

X → f_{∗}f^{∗}X

is a weak equivalence.

Proof: Let J/U be the category of J-objects over U. An object of J/U is a pair (V, i) consisting of an object V of J and a map i : fV → U. A map from (V_{0}, i_{0}) to (V_{1}, i_{1}) is a map g : V_{0} → V_{1} with the property that i_{1} ∘ f(g) = i_{0}. There is an obvious functor

p : J/U → J

given by p(V, i) = V.

For a diagram Y : J → A_{∞}, the diagram f_{∗}Y is given by

f_{∗}Y (U) = \underleftarrow{lim}_{J/U}p^{∗}Y.

If the diagram Y is fibrant, so is is p^{∗}Y. This means that the map from ho \underleftarrow{lim} p^{∗}Y → \underleftarrow{lim} p^{∗}Y is a weak equivalence. We can therefore study f_{∗}Y (U) using the homotopy spectral sequence. Now this spectral sequence takes the form

\underleftarrow{lim} _{J/U}^{s}π_{t}p^{∗}Y ⟹ π_{t−s}f_{∗}Y (U).

Since the diagram X is a realization of D, the groups π_{2t+1}p^{∗}Y (V) are zero, and π_{2t}p^{∗}Y (V) is the module of global sections of ω^{t}. Since the funcotr “pullback,” described below, is an equivalence of categories, the inverse limit can be computes over Et(J)/U and so coincides with the Čech cohomology H^{s}(ob J; ω^{t}). But since U is affine, and ω is quasicoherent, these cohomology groups are zero, unless s = 0, in which case it is the module of global sections of ω over U. The spectral sequence therefore degenerates to its edge homomorphism. This completes the proof.

We have used

Lemma F.2. The forgetful functor F : J/U → Et(J)/U, and the functor “pullback”

P : Et /U → J/U
(i : V → U) ↦ i^{∗}A_{U}

are inverse natural equivalences.

Proof: The composite P F is the identity. The composite P F sends the data

i : V → U t : A_{V} ⟶^{≈} i^{∗}A_{U}

to the data

i : V → U 1 : i^{∗}A_{U} → i^{∗}A_{U}.

The map t defines the natural isomorphism from P F to the identity functor.

Working remark 32. The hypotheses on the subcategory J are needed in order that the Čech cohomology and the derived functors of inverse limit coincide. Make this lemma explicit. Also, this lemma is a general one about stacks. It is the whole reason that a “representable” stack is equivalent to a scheme.

Working remark 33. OK, what do I mean by fibrant? I want the inverse limit to be the homotopy inverse limit. I think this means I mean fibrant in the model category where weak equivalences and cofibrations are given termwise. I think this means that for each object U of I, the map

X(U) → \underleftarrow{lim}_{(I/U)′}X

ELLIPTIC COHOMOLOGY (February 22, 1999) 83

is a fibration. I hope things work out so that if X is a “fibrant diagram, then so is f^{∗}X.

Now to compare the realization spaces. There are a few realization spaces that need to be considered. First there is the space rD. Next there is the space r̃D which can be defined either as the nerve of the full subcategory of A_{∞}^{I} with objects the diagrams which admit an isomorphism in ho A_{∞} with D, or as the category whose objects are diagrams X together with an isomorphism in ho A_{∞} with D, and in which a morphism consists of a weak equivalence f : X_{0} → X_{1}, and an automorphism t : D → D such that

ho X_{0} ⟶^{ho f} ho X_{0}
↓  ↓
D ⟶_{t} D

commutes. The first description is easier to work with, while the second makes it clear that there is a fibration

rX → r̃X → B Aut D,

so it is only the higher homotopy groups of the two spaces that differ.

Lemma F.3. The map

r̃D → r̃f^{∗}D

is a weak equivalence.

Proof: We are interested in comparing certain components of classifying spaces of weak equivalences between the categories A_{∞}^{I} and A_{∞}^{J}. These can be calculated as the classifying spaces of weak equivalences between cofibrant-fibrant objects in any model category structure. The result therefore reduces to the lemma above.

Lemma F.4. The map Aut D → Aut f^{∗}D is an isomorphism.

Proof: The diagrams D and f^{∗}D are sections of

ho A_{∞}^{Ell} → I

and it’s restriction to J. The groups Aut D and Aut f^{∗}D thus sit in exact sequences

Aut^{v} D → Aut D → Aut 1_{I}
Aut^{v} f^{∗}D → Aut f^{∗}D → Aut 1_{J},

where the superscript “v” indicates “vertical.” Let A be the abelian group valued functor (on D) with

A(U) = ho A_{∞}^{Ell}(E(U), E(U))_{1}.

Then by Proposition A.6, there are natural isomorphisms

Aut^{v} D ≈ \underleftarrow{lim} A
Aut^{v} f ∗ D ≈ \underleftarrow{lim} f^{∗}A.

But the functor A is a sheaf, and J is a cover of the terminal object. Both limits are therefore equal to H^{0}(A). Finally, the map

Aut 1_{I} → Aut 1_{J}

is an isomorphism, since the functor “isomorphisms between elliptic curves” is a sheaf. This completes the proof.

Corollary F.5. The map

rD → rf^{∗}D

is a weak equivalence.

Now take J to be the full subcategory with underlying schemes whih are spectra of local rings. I think I can do this. In this case the input of the spectral sequence involves just sheaf cohomology (as opposed to Čech cohomology of some specific complex). I think this makes life easier, but I am not sure. This is related to the Godemont resolution.

References

1. J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, 1974.

2. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, no. 304, Springer–Verlag, Berlin, 1972.

3. P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular Functions of One Variable II, Lecture Notes in Mathematics, no. 349, Springer–Verlag, 1973, pp. 143–316.

ELLIPTIC COHOMOLOGY (February 22, 1999) 85

4. W. G. Dwyer and D. M. Kan, Centric maps and realization of diagrams in the homotopy category, Proceedings of the American Mathematical Society 114 (1992), 575–584.

5. W. G. Dwyer, D. M. Kan, and C. R. Stover, An e^{2} model category structure for pointed simplicial spaces, Journal of Pure and Applied Algebra 90 (1993), 137–152.

6. D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer–Verlag, New York, 1972.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

E-mail address: mjh@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

E-mail address: hrm@math.mit.edu