Lecture:

Spletna Učilnica

Description

This course is an introduction to commutative algebra.

Contents

Week 1: Rings and Modules Recap

A brief repetition of concepts from ring and module theory, in particular in the commutative setting (including tensor products of modules). Assuming you have taken Algebra 3, a few things may still be new: the Prime Avoidance Lemma, the Nakayama Lemma, Hom-Tensor adjunction and left exactness of Hom-functors and right exactness of the tensor product. (Chapters 1.1-1.3 of [AF23]).

Notes 17.2.

Recap: Rings, Ideals, Prime and Maximal Ideals, Radicals, Nil- and Jacobson Radical, Prime Avoidance Lemma, Quotients of Rings by Ideals (Universal Property and Isomorphism Theorems).

Corresponding to Chapters 1.1 and 1.2 of [AF23].

Notes 20.2.

Recap: Chinese Remainder Theorem for Rings, Modules, Quotients, Isomorphism Theorems, Products and Direct Sums.

Corresponding to p.12-16 (parts of Chapters 1.2, 1.3) and p.20 (part of Chapter 1.4) of [AF23].

Week 2: Modules Recap, Free and Projective Modules

Notes 24.2.

Free modules, Finitely Generated Modules, Nakayama Lemma, Exact Sequences, Tensor Products, Left Exactness of Hom, Hom-Tensor Adjunction, Right Exactness of Tensor Products.

Corresponding to most of the remaining parts of Chapters 1.3 and 1.4 of [AF23].

Notes 27.2.

Rank of free modules; failure of exactness of tensor product and Hom-functors; projective modules.

Chapter 1.4 of [AF23]. Projective modules only appear in an exercise (Ferretti has a different book with an introduction to homological algebra). More information on projective modules can be found in [C24, Chapter 3.5] or in [H74, Chapter IV.3] (and many other algebra books).

Week 3: Injective and Flat Modules, Localization

Notes 3.3.

Characterization of Injective Modules (including Baer's Criterion); Characterization of Flat Modules.

A reference for the characterization of injective modules is [C24, Section 3.6]. For flat modules, [S, Lemma 10.39.5] does the proof in a way close to the notes, but they invoke that right exact functors preserve colimits to reduce to the finitely generated case. When considering \(M \subseteq N\), the reduction to \(M\) finitely generated is elementary, as in Obs.1 in the notes. Reducing to \(N\) finitely generated is trickier. One either needs to work with colimits or appeal to the actual construction of the tensor product to see this. In the notes this reduction is avoided.

There also exists a (very different) proof of Theorem 1.18 (e)=>(b) which uses a duality \(M \mapsto \operatorname{Hom}_{\mathbb Z}(M, \mathbb Q/\mathbb Z)\) to reduce it to Baer's criterion for injectivity. See Lam's Lectures on Modules and Rings, Chapter 2, paragraph (4.12) for a nice presentation of this.

A proof of the (key part of the) Snake Lemma appears in the movie "It's My Turn".

Notes 6.3.

Some concluding observations on Flat modules. Localization of Rings (starting Chapter 1.6 of [AF23]).

Week 4: Localization

Notes 10.3.

Localization of Rings and Modules; Algebras; Extension and Contraction of Ideals; Spectrum; Nilradical as Intersection of Prime Ideals; Exactness of Localization.

Chapter 1.6 of [AF23] (plus some additions).

Notes 13.3.

Residue Fields; Local Properties; Extension of Scalars and Localization of Modules as Extension of Scalars.

Chapter 1.6 of [AF23] plus some additions.

Week 5: Tensor Products of Algebras, Finitely Generated Modules over PIDs

Notes 17.3.

Localization of Tensor Products; Tensor Products of Algebras; Recap PIDs; Submodules of free modules are free over a PID; Smith Normal Form; Stacked Bases.

Chapters 1.6, 1.5, 2.1, 3.1 in [AF23].

Notes 20.3.

Structure Theorem for finitely generated modules over PIDs.

Chapters 1.5 and 2.1 in [AF23], plus uniqueness.

Week 6: Torsion, Noetherian/Artinian Modules and Rings, Minimal Primes

Notes 24.3.

Torsion Modules. Noetherian/Artinian Modules and Rings. Hilbert's Basis Theorem.

[AF23, Chapters 2.1-2.3]

Notes 27.3.

For noetherian rings: "Weak" local-global result. Finiteness of minimal primes; nilpotency of nilradical. Artinian rings: finiteness of spectrum; nilpotency of nilradical.

Chapters 2.3, 2.4 of [AF23].

Week 7: Artinian Rings, Primary Decomposition

Notes 31.3.

Structure Theory of Artinian Rings; Primary Ideals, Primary Decomposition.

Chapters 2.4 and 3.2 of [AF23]

Notes 3.4.

First and Second Uniqueness Theorem for Primary Decompositions.

Chapter 3.2 of [AF23].

Week 8: Primary Decomposition, Integral Ring Extensions, Cohen-Seidenberg Theorems

Notes 7.4.

Applications of Primary Decomposition; Integral Ring extensions.

Chapters 3.2 and 5.1 of [AF23].

Notes 10.4.

Integral extensions: Lying Over, Incomparability, Going Up

Chapter 5.2 in [AF23].

Week 9: Going Down, Noether Normalization, Dimension, Affine Varieties

Notes 14.4.

Going Down Theorem; Noether Normalization and Zariski's Lemma; Dimension of Polynomial Rings; Relation to Transcendence Degree;

Chapters 5.2, 5.3 in [AF23]. For dimension, Chapter 11 of [G13].

Notes 17.4.

Final remarks about Krull dimension; Affine Varieties.

Chapters 11, 10 in [G13], Chapter 8.1 in [AF23].

Week 10: Hilbert's Nullstellensatz

Notes 24.4.

Hilbert's Nullstellensatz in various forms.

Chapter 8.2 in [AF23]. Chapter 10 in [G13].

Week 11: Irreducible Varieties, Morphisms, Zariski Tangent Space

Notes 28.4.

Noetherian Topological Spaces, Irreducible Varieties, Decomposition into Irreducible Components; Morphisms of Varieties, Category anti-equivalence with Finitely Generated Reduced Algebras in the algebraically closed case. Examples.

Notes 30.4.

Noether Normalization and Localization in the Geometric Context; Tangent Spaces of Affine Varieties (geometrically). Zariski (Co)Tangent space of a Regular Local Ring.

Ch. 8.5, 8.8 of [AF23], Ch. 11 [G13].

Week 12: Regularity, Krull's Principal Ideal Theorem, DVRs`

Notes 12.5.

Regular Local Rings; Krull's Principal Ideal Theorem; Discrete Valuations and DVRs.

[G13, Ch. 11 and 12] [AF23, Ch. 7.3]

Notes 15.5.

Characterizations of DVRs.

[AF23, Ch. 73], [G13, Ch. 12]

Week 13: Dedekind Domains, Class Group

Notes 19.5.

Dedekind domains, Prime Factorization of Ideals, Fractional and Invertible Ideals.

[G13, Ch.13], [AF23, Ch 3.4]

Notes 22.5.

The group of invertible fractional ideals, operations on ideals, class groups.

[G13, Ch. 13]

Week 14: Inverse Limits, Topological Groups, Group Completions

Notes 26.5.

Inverse Systems and Inverse Limits of Groups/Rings/Modules; Topological Groups

[AF23, Ch. 7.5], [AM69, Ch. 10]

Notes 29.5.

Completion of first countable topological groups using Cauchy sequences; linear topologies; connection between Cauchy sequence and the inverse limit definition; exactness properties.

[AF23, Ch. 7.5], [AM69, Ch. 10]

Week 15: Completions of Rings and Modules

Notes 2.6.

I-adic topology; Artin-Rees Lemma; Krull's Intersection Theorem.

[AF23, Ch. 7.5], [AM69, Ch. 10]

Bibliography

Here is a list of books that can be useful as supporting materials.

Main Books

[AF23]

A. Ferretti. Commutative Algebra. Grad. Stud. Math., 233, American Mathematical Society, Providence, RI, 2023.

[AM69]

M.F. Atiyah and I.G. Macdonald. Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.

Additional Books

[AK21]

A. Altman, S. Kleiman. A term of Commutative Algebra. Version from 2021-04-11. Available online for free, e.g. dspace.mit.edu.

[B72]

N. Bourbaki. Elements of mathematics. Commutative algebra. Translated from the French. Hermann, Paris; Addison-Wesley Publishing Co., Reading, MA, 1972.

[E95]

D. Eisenbud. Commutative Algebra (with a View Toward Algebraic Geometry). 1995.

[H74]

T.W. Hungerford. Algebra. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1974.

[M80]

H. Matsumura. Commutative algebra. Second edition. Math. Lecture Note Ser., 56, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1980.

[M89]

H. Matsumura. Commutative ring theory. Cambridge Stud. Adv. Math., 8, Cambridge University Press, Cambridge, 1989.

[S00]

R. Y. Sharp. Steps in commutative algebra. Second edition. London Math. Soc. Stud. Texts, 51, Cambridge University Press, Cambridge, 2000.

[ZS75]

O. Zariski, P. Samuel. Commutative algebra. Vol. 1. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958 edition. Grad. Texts in Math., No. 28, Springer-Verlag, New York-Heidelberg-Berlin, 1975.

[ZS60]

O. Zariski, P. Samuel. Commutative algebra. Vol. II. Univ. Ser. Higher Math. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960.

Lecture Notes

[G13]

A. Gathmann. Commutative Algebra. 2013/14. Link to PDF.

[C24]

P. L. Clark. Commutative Algebra. 2024. Link toPDF.

[S]

The Stacks Project. stacks.math.columbia.edu.