5/4/20: Homework 7 is due at the beginning of class. A criterion for a
complex to be a resolution. [Class notes. Eisenbud: Section
20.3 through the proof of Theorem 20.9]
5/4/20: Homework 7 is due at the beginning of class. A criterion for a
complex to be a resolution. [Class notes. Eisenbud: Section
20.3 through the proof of Theorem 20.9]
5/1/20: Fitting ideals and projective modules of constant rank. Depth
and Ext. [Class notes. Eisenbud: Sections 20.2 (Proposition
20.8) and 18.1.1]
4/29/20: Fitting ideals: base change, localization, obstruction to
generation by j elements, relation to annihilators. [Class
notes. Eisenbud: Section 20.2 (through Proposition 20.7)]
4/27/20: Uniqueness of a minimal free resolution revisited: the proof
of main theorem (20.2). Fitting ideals: definition. [Class
notes. Eisenbud: Sections 20.1-20.2 (through Corollary 20.4)]
4/24/20: More implications of being CM. Uniqueness of a minimal free
resolution revisited. [Class notes. Eisenbud: Sections 18.2
(through the end) and 20.1 (before the proof of 20.2)]
4/20/20: Homework 6 is due at the beginning of
class. Depth. Cohen-Macaulay rings. [Class notes. Eisenbud:
Sections 18.1 (skipping Claims 18.2-18.3 and 18.1.1 for the time
being) and 18.2 (just the definitions and wording of theorems through
Proposition 18.8 for the time being)]
4/17/20: Homological theory of regular local rings: projective and
global dimension. [Class notes. Eisenbud: Sections 19.1-19.3]
4/15/20: Koszul complexes and depth. [Class notes. Eisenbud:
Section 17.3 (the rest)]
4/13/20: Koszul complexes of arbitrary length. Building the Koszul
complex from parts. [Class notes. Eisenbud: Sections 17.2-17.3
(through Corollary 17.11, skipping Corollary 17.10 for the time being]
4/10/20: Koszul complexes of length 1 and 2. [Class
notes. Eisenbud: Section 17.1]
4/8/20: Minimal free resolutions. [Class notes. Eisenbud:
Section 19.1]
4/6/20: Homework 5 is due at the beginning of class. Buchberger's
algorithm. [Class notes. Eisenbud: Section 15.4]
4/3/20: The existence of Gröbner bases. Group work (get a paper
and pen/pencil ready). [Class notes. Eisenbud: Section 15.2
(from Gröbner bases). Cox, Little, O'Shea: Sections
2.5-2.7]
4/1/20: The division algorithm. Initial ideals. Gröbner
bases. [Class notes. Eisenbud: Sections 15.2 and 15.3]
3/30/20: Review: graded rings. Monomial orders. [Class
notes. Eisenbud: Sections 1.9, 15.1, and 15.2 (through weight
orders)]
3/27/20: Catenary rings. The weak Nullstellensatz. Finiteness of the
integral closure. [Class notes. Eisenbud: Sections 13.1
(Corollary 13.6), 13.2, and 13.3 (Corollary 13.13)]
3/25/20: Strong Noether normalization. Dimension and transcendence
degree. Going down for integral extensions of normal rings. [Class
notes. Eisenbud: Section 13.1 (Theorems 13.3, A, Corollary 13.4,
Proposition 13.10, and Theorem 13.9)]
3/23/20: Homework 4 is due at the beginning of class. Dedekind
domains. Noether normalization: another proof of the dimension axiom
D4 and weak Noether normalization. [Class notes. Eisenbud:
Sections 11.4 (skip Theorem 11.8), 13.1 (through Lemma 13.2), and
8.2.1 (Theorem A1)]
3/20/20: Invertible modules, the Picard group, and Cartier
divisors. [Class notes. Eisenbud: Section 11.3]
3/18/20: Recollection of what we learned about normal rings before the
break. The proof of Serre's criterion. [Class notes. Eisenbud:
Section 11.2 (proof of Theorem 11.5)]
3/9-17/20: Extended Spring Break. [Have a good rest and study what
we have done and do the homework]
3/6/20: A normal Noetherian domains is the intersection of its
localizations at codim-1 primes. Geometric interpretation. Serre's
criterion for the normality of a Noetherian ring (wording). A
Noetherian ring is normal iff it is a finite product of normal
domains. The analogue of Serre's criterion for a Noetherian ring to be
reduced. [Class notes. Eisenbud: Section 11.2 from Corollary
11.4 up to the proof of Theorem 11.5]
3/4/20: Class canceled. I am out of town. [Study what we have done
and start doing the homework]
3/2/20: Normal rings, continued. [Class notes. Eisenbud:
Section 11.2 up to Corollary 11.4]
2/28/20: Homework 3 is due at the beginning of class. Complete regular
local rings. DVRs (home reading). Normal rings and normalization:
recollection and introduction. [Class notes. Eisenbud: Sections
10.3 (Corollary 10.16), 11.1 (read on your own) and 11.2 (first page)]
2/26/20: Regular local rings. [Class notes. Eisenbud: Section
10.3 through Corollary 10.15]
2/24/20: Verification of dimension axiom D4 (finished). Reminder on
Nakayama's lemma. [Class notes. Eisenbud: Section 10.2 (the
proof of Corollary 10.13)]
2/21/20: Proofs of the Base-Fiber theorem (10.10) and some idea on the
proof of Lemma 10.11. Verification of dimension axioms D1 (the second
half) and D4 (just the wording). [Class notes. Eisenbud:
Section 10.2 through the wording of Corollary 10.13]
2/19/20: Systems of parameters. Dimension of base and fiber (no proofs
yet): geometric interpretation and examples. [Class
notes. Eisenbud: Sections 10.1 and 10.2 through Lemma 10.11
(skipping the proofs)]
2/17/20: Comments on symbolic powers. Some consequences of the
Principal Ideal theorem, including its own converse. [Class
notes. Eisenbud: Section 10.0]
2/14/20: Homework 2 is due at the beginning of class. Principal Ideal
theorem: two versions. [Class notes. Eisenbud: Section 10.0
through Theorem 10.2]
2/12/20: Relative dimension zero: verifying Axiom D3. Principal prime
ideals have codimension at most 1. [Class notes. Eisenbud:
Sections 9.1 and 10.0 (the first page)]
2/10/20: Dimension theory. [Class notes. Eisenbud: Sections 8.1
and 9.0]
2/7/20: Example: the homology of tensor product of complexes over a
field (algebraic Künneth formula). Tor (M,N) may be computed by
resolving either M or N. [Class notes. Eisenbud: Section
A3.13.4 (i. Balanced Tor)]
2/5/20: The spectral sequence(s) of a double complex. [Class
notes. Eisenbud: Section A3.13.4 up to i. Balanced Tor]
2/3/20: The spectral sequence of a filtered complex. The general
definition of a spectral sequence and its convergence. Two filtered
complexes associated to a double complex. [Class
notes. Eisenbud: Section A3.13.3 up to Theorem A3.22]
1/31/20: Homework is due at the beginning of class. The spectral
sequence of a filtered complex: the first two terms and the spectral
sequence of a subcomplex. [Class notes. Eisenbud: Section
A3.13.3 through page 662 (We have taken a different approach in class,
though)]
1/29/20: Exactness of right-exact functors on projectives and
injectives. Local cohomology. The spectral sequence of a filtered
complex: introduction. [Class notes. Eisenbud: Sections
A3.11.2, A3.13.1 (an alternative introduction to the one we discussed
in class with the LES coming from a subcomplex), and A3.13.3
before 1E on page 662]
1/27/20: Derived functors. Tor and Ext. [Class notes. Eisenbud:
Sections A3.9-A3.11.1]
1/24/20: Long exact sequence. The horseshoe lemma. [Class
notes. Eisenbud: Section A3.8]
1/22/20: Introduction. Syllabus handed out. Overview of the
course. Homology and projective resolutions. The snake
lemma. [Syllabus. Class
notes. Eisenbud: Sections A3.5, A3.6 (especially Corollary
A3.14b), A3.7]
Last modified: (2020-09-09 02:36:46 CDT)