12/11/19: The last homework is due at the beginning of class.
Homology of complexes. Maps of complexes and homotopies. The existence
and uniqueness of a projective resolution. [Class
notes. Eisenbud: Sections A3.5-A3.6 through Corollary A3.14a]
12/11/19: The last homework is due at the beginning of class.
Homology of complexes. Maps of complexes and homotopies. The existence
and uniqueness of a projective resolution. [Class
notes. Eisenbud: Sections A3.5-A3.6 through Corollary A3.14a]
12/9/19: Injective envelopes (hulls) and minimal injective
resolutions. [Class notes. Eisenbud: Section A3.4 (after the
example of an injective resolution of Z)]
12/6/19: Injective Z-modules. Injective resolutions. [Class
notes. Eisenbud: Section A3.4 through the example of an injective
resolution of Z after Corollary A3.9. See also
a Clarification on the proof of Lemma A3.8]
12/4/19: Projective modules. Projective resolutions. Injective
modules. Baer's criterion for injectivity. [Class
notes. Eisenbud: Sections A3.2 (through Proposition A3.1), A3.3,
and A3.4 through Lemma A3.4]
12/2/19: Homework is due at the beginning of class. Proofs of main
results from Chapter 7: Theorem 7.16 (b, c), Theorem 7.7 (Cohen's
structure theorem), and Corollary 7.17 (an Inverse function
theorem). [Class notes. Eisenbud: Section 7.6: after Theorem
7.16 through Corollary 7.17; proof of Cohen's structure theorem 7.7
modulo the existence of coefficient field; read the proofs of
Corollary 7.17 and Hensels' lemma (Theorem 7.3) on your own]
11/29/19: Thanksgiving Break. Have a happy Thanksgiving!
11/27/19: Snow day: no classes. More time to think about life and
homework!
11/25/19: Hensel's lemma. A version of the implicit function
theorem. Cohen's structure theory. Maps from power series
rings. [Class notes. Eisenbud: Sections 7.2 (from Theorem 7.3),
7.4 (Theorem 7.7), and 7.6 (through Theorem 7.16)]
11/22/19: Completions: Noetherianness and flatness. [Class
notes. Eisenbud: Sections 7.2 (through Theorem 7.2), and 7.5
(Proposition 7.12, Corollary 7.13, and proofs of Theorems 7.1 and
7.2)]
11/20/19: Direct limits: homework-style example. Inverse limits.
Examples. Completions: definitions. [Class
notes. Atiyah-MacDonald: the Topology and Completions section from
Chapter 10. Eisenbud: Section 7.1]
11/18/19: Flatness criteria, continued. Direct limits. [Class
notes. Eisenbud: Section 6.3 (Corollary 6.3, study Lemma 6.4 and
Corollary 6.5 on your own). Atiyah-MacDonald: Exercises
2.14-2.19]
11/15/19: Homework is due at the beginning of
class. Tor. Flatness. [Class notes. Eisenbud: Sections 6.2 and
6.3 through Proposition 6.1]
11/13/19: The Hilbert syzygy theorem. Flat
families. Examples. [Class notes. Eisenbud: Section 1.10
and Chapter 6 through Section 6.1]
11/11/19: An application of the Krull intersection
theorem. Counterexamples. Free resolutions. Examples. [Class
notes. Eisenbud: Sections 5.3 (from Corollary 5.5) and 1.10 (up to
Theorem 1.13)]
11/8/19: Using the blowup algebra to prove the Artin-Rees lemma. The
Krull intersection theorem. [Class notes. Eisenbud: Section 5.2
(from Prop. 5.3), Corollary 4.7 (of the Cayley-Hamilton theorem) and
Section 5.3 through Corollary 5.4]
11/6/19: Stability and finiteness. The initial form of an element. The
blowup algebra. [Class notes. Eisenbud: Sections 5.1-5.2 before
Prop. 5.3]
11/4/19: Proof of the Hilbert polynomial theorem. Filtrations. The
associated graded ring. [Class notes. Eisenbud: Section 1.9
(after Theorem 1.11), Introduction to Chapter 5, and Section 5.1 (the
first page)]
11/1/19: Homework is due at the beginning of class. Deduction of
Theorem 1.6 from the general form of Nullstellensatz. Graded rings and
Hilbert polynomials. [Class notes. Eisenbud: Theorem 1.6 from
Section 4.5 and Sections 1.5 and 1.9 through Theorem 1.11]
10/30/19: Primes in an integral extension, continued. The
Nullstellensatz (a stronger form). Deduction of Corollary 1.9 from
it. [Class notes. Eisenbud: Sections 4.4 (Proposition 4.15 and
Corollary 4.18) and 4.5 through Corollary 1.9]
10/28/19: Every UFD is normal. Normalization commutes with
localization. Primes in an integral extension: going up and
down. Geometric examples. [Class notes. Eisenbud: Sections 4.2,
4.3 (read on your own), and the first two paragraphs of 4.4]
10/25/19: Normalization. Integrally closed subrings and integral
extensions of rings. Normal domains and rings. Geometric
interpretation (study at home). [Class notes. Eisenbud:
Introduction to Chapter 4]
10/23/19: Geometric interpretation of primary decomposition (study at
home). Nakayama's Lemma and the Cayley-Hamilton theorem. [Class
notes. Eisenbud: Sections 3.8 (read on your own) and 4.1]
10/21/19: The second uniqueness statement for primary
decomposition. Irreducible ideals and the existence of primary
decomposition in Noetherian rings. Another characterization
of m-primary ideals. [Class notes. Atiyah-MacDonald:
Corollary 4.11, Sections 7.11-7.17]
10/18/19: Homework is due at the beginning of class. Minimal primes of
an ideal. Primary ideals and localization. Primary decomposition and
localization. [Class notes. Atiyah-MacDonald: Proposition 4.6
through Proposition 4.9]
10/16/19: The proof of the first uniqueness theorem. Minimal primary
decomposition and the set of associated primes of an ideal. [Class
notes. Atiyah-MacDonald: Theorem 4.5 up to Proposition
4.6. Eisenbud: Theorem 3.10b (ideal case only) ]
10/14/19: Primary decomposition of an ideal. The first uniqueness
theorem for primary decomposition (no proof yet). [Class
notes. Atiyah-MacDonald: Proposition 4.2 through Theorem 4.5
(before the proof)]
10/11/19: Characterization of primary ideals. The radical of a primary
ideal. [Class notes. Eisenbud: Section 3.3 through Proposition
3.9 (ideal case only). Atiyah-MacDonald: Chapter 4 up to
Prop. 4.2]
10/9/19: Proof of Theorem 3.1, concluded. Primary ideals and primary
submodules. The definition of p-primary ideals. The radical of a ring
and prime ideals. [Class notes. Eisenbud: Section 3.2 through
the end and the first two paragraphs of Section 3.3, Corollary 2.12]
10/7/19: Prime avoidance, continued: finishing the proof of Lemma
3.3. Proof of Theorem 3.1, continued. [Class notes. Eisenbud:
The Proof of Case 2 of Lemma 3.3, Corollaries 3.2 and 3.5, and Lemma
3.6]
10/4/19: The set of associated primes. Proof of Part (b) of Theorem
3.1. Prime avoidance. [Class notes. Eisenbud: Theorem 3.1,
Proposition 3.4, Remark after Corollary 3.5, Lemma 3.3 (the proof of
Case 1 so far)]
10/2/19: Homework is due at the beginning of class. Associated primes:
motivational examples and definition. [Class notes. Eisenbud:
Introduction to Chapter 3, Section 3.1 (before Theorem 3.1)]
9/30/19: Noetherian rings. The Hilbert basis theorem
(proof). Corollaries. [Class notes. Eisenbud: Section 1.4 up to
1.4.1 (with proofs)]
9/27/19: Artinian rings are Noetherian. More on the structure of
Artinian rings. The Hilbert basis theorem (wording). [Class
notes. Eisenbud: Section 2.4 through the end and Section 1.4 up to
1.4.1 (with no proofs so far)]
9/25/19: The structure of finite-length modules via their
localizations. [Class notes. Eisenbud: Theorems 2.13(b,c)]
9/23/19: Finite-length modules are the same as those which are
Noetherian and Artinian. ACC and DCC for vector spaces. The structure
of finite-length modules via their localizations. [Class
notes. Eisenbud: Theorem 2.13(b), just the wording so far]
9/20/19: Noetherian and Artinian rings. Composition series,
length. Finite-length modules. [Class notes. Eisenbud: Section
2.4 through Theorem 2.13(a), including the proof]
9/18/19: Homework is due at the beginning of class. Noetherian and
Artinian modules and rings. [Class notes. Eisenbud: Section 1.4
(Definitions only so far) skipping 1.4.1]
9/16/19: The geometric meaning of localization. The support of a
module. Localization to maximal ideals. Closed monoidal structure or
the adjointness of Hom and tensor. [Class notes. Eisenbud:
Section 2.2 from Corollary 2.7]
9/13/19: Local rings and Rp, where p is a prime ideal. Hom
and tensor product. [Class notes. Eisenbud: Section 2.2
through Proposition 2.5]
9/11/19: The geometric meaning of ideals, prime ideals, and maximal
ideals. Ideal theory in R[U-1]. [Class
notes. Eisenbud: Corollary 1.9 and Proposition 2.2]
9/9/19: Localization. [Class notes. Eisenbud: Section 2.1
before Proposition 2.2]
9/6/19: Syllabus handed out. Some motivating questions for the
course. Modules and related terminology: annihilator, faithful
modules, direct sums and products, exact sequences, finite generation
and
presentation. [Syllabus. Class
notes. Eisenbud: Section 0.3]
9/4/19: Introduction. Overview. Connections to commutative algebra. A
crash course in affine algebraic
geometry. Hilbert's Nullstellensatz.
Class notes. Eisenbud: Sections 1.1-1.3, 1.6]
Last modified: (2020-01-16 22:34:39 CST)