Commutative Algebra — lectures winter semester 2019/20. Usosweb page Exercise session - in particular homework assignments Office hours: Wed 10-12. Please send an email first!

Description of lectures

1. I Lecture, 2.10:
   • preliminaries: definition of k-algebras and their homomorphisms. Domains and prime ideals, fields and maximal ideals. Extended example: maximal ideals in C(S^1, RR) as points of S^1 and topology.
   • Spec: definition of Spec as set, check that f:A->B induces f*:Spec(B)->Spec(A), definition of Zariski topology and check that it indeed is a topology. Check that f^* is continuous. Example: Spec(CC[x]).
   • Announcement: weak Nullstellensatz. Drawing Spec_{max}(k[x_1..x_n]/I) as subsets of k^n.
   • Localization in S: definition by first dividing by elements annihilated by something in S and then taking the usual construction in the quotient ring.
2. II Lecture, 9.10:
   • Universal property of localization. Localization at f is equal to A[x]/(xf-1).
   • Properties of localization: it is zero iff S contains zero, A -> S^{-1}A is injective iff S consists of nonzerodivisors, A -> S^{-1}A is an isomorphism iff S consists of invertible elements.
   • Every ideal in localization is extended from A. Prime ideals in localization = prime ideals that do not intersect S. Definition and characterization of nilradical.
   • Basics of modules: definition, homomorphisms, free and finitely generated modules. Examples of A-modules: A/I, I, A-algebras. Cayley-Hamilton trick formulated.
3. III lecture, 16.10:
   • Cayley-Hamilton trick -- proof. Nakayama lemma, local Nakayama, M = N + mM implies M = N, generators of M/mM lift to generators of M for M fin.gen.
   • Tensor product: universal property, uniqueness up to iso and construction. Adjoint functors (info only).
   • Modules of differentials: universal property and construction. Differentials of polynomial rings over a field.
4. IV lecture, 23.10:
   • Rank of free module. Cotangent module for a polynomial ring k[x_1,...,x_n] is free of rank n. (*Definition of a smooth algebra — auxiliary information).
   • Exactness and short exact sequences. The tensor product is right-exact.
   • Localization of a module. S^{-1}A \tensor_A M = S^{-1}M. A/I\tensor_A M = M/IM.
   • Tensor product of A-module with A-algebra B is a B-module. Tensor product of two A-algebras is an A-algebra and in fact a coproduct. Examples: tensor product of polynomial rings is a polynomial ring, tensor product of A[x_1..x_s]/I and A[y_1..y_t] is A[x_1,...,x_s, y_1,...,y_t]/((I) +(J)) — without proof.
   • Residual field kappa(pp) at pp\in Spec(A). Fiber of a morphism is f:A->B is kappa(pp) \tensor_A B and Spec(fiber) = fiber of f^* over pp.
5. V lecture, 30.10:
   • Noetherian modules, behaviour under short exact sequences.
   • Noetherian rings. Finitely generated modules over Noetherian rings are Noetherian. Localizations and quotients of Noetherian rings are Noetherian. Finitely generated algebras over Noetherian rings are Noetherian (main workhorse: Hilbert basis theorem).
   • Finite and integral ring homomorphisms. Finite implies integral, finitely generated algebras with integral generators are finite. Examples. Integral closure and normalization. Integral closure is a ring. Without proof: closure of ZZ in finite extension of QQ is Noetherian, for an integral domain A that is finitely generated as a k-algebra the normalization is a finitely generated k-algebra as well and it is even finite over A.
6. VI lecture, 6.11:
   • In an integral extension of domains A->B we have A a field iff B is a field.
   • Integral extensions and prime ideals: incomparability of ideals in fiber, Spec(B)->Spec(A) is surjective for A->B an integral injective homomorphism.
   • Krull dimension. Fields have dimension zero, PIDs have dimension one, fibers of finite homomorphisms have dimension ≤ 0. If A -> B is an integral injective homomorphism, then dim(B) = dim(A).
   • Nagata's trick and dim k[x_1,...,x_n] = n.
7. VII lecture, 13.11: midterm exam.
8. VIII lecture, 20.11:
   • Reminder on finite and integral extensions, dim k[x_1,...,x_d]/I < d for I nontrivial and dim k[x_1,...,x_d]/(f) = d-1.
   • Noether normalization, strong version ([13.3, Eisenbud] but with prime ideals). Corollary: weak Nullstellensatz.
   • Strong Nullstellensatz.
9. IX lecture, 27.11:
   • Algebraic sets: definitions, Zariski topology. (today kk algebraically closed)
   • Nullstellensatz, geometrically: algebraic sets in bijection with radical ideals of kk[x_1,...,x_n]. Irreducibility, irreducible algebraic sets in bijection with prime ideals.
   • Each algebraic sets decomposes into a union of finitely many irreducible ones. Each radical ideal is an intersection of finitely many prime ideals.
   • Classification of prime ideals in kk[x_1,...,x_n] for n ≤ 2.
   • Other ways of expressing dimension of fin.gen. kk-algebras A that are domains: transcendence degree of fraction field. This degree is equal to the dimension of A.
   • Bonus (will not appear on exams): Spec(QQbar \otimes_QQ QQbar) is in bijection with Gal(QQbar/QQ) and QQbar \otimes_QQ QQbar is not Noetherian.
10. X lecture, 4.12:
    • Discrete valuation rings: definitions, ideals in a DVR
    • DVRs are exactly Noetherian normal local one-dimensional domains,
    • Localization of normal domain is a normal domain. Localization of a normal Noetherian domain A at a prime pp with dim(A_pp) = 1 is a DVR. Dedekind domains: definition and localization at maximal ideals is a DVR.
11. XI lecture, 11.12:
    • A domain is the intersection of all its localizations at maximal ideals. Corollary: noetherian domains having DVRs as localizations at maximal ideals are Dedekind domains.
    • Local properties: being zero, being a monomorphism/epimorphism/isomorphism of modules is a local property (see Atiyah-Macdonald p.30).
    • Every ideal in a Dedekind domain is a product of prime ideals. Every ideal in a Dedekind domain is generated by at most two elements (proof had an issue, see corrected proof. I put also the TeX source file).
    • Local theory: Krull's principal ideal theorem and corollary: dim(A_mm) ≤ dim(mm/mm^2); so far without proofs. Regular local rings. Regular local rings are UFDs (without proof and we won't give one).
12. XII lecture, 18.12:
    • Example: application of Jacobian criterion to prove that y^2 = x^3 + x is regular.
    • Proofs of Krull's principal ideal theorem and corollaries.
    • The set of regular points is open and non-empty in a finitely generated kk-algebra that is a domain (char kk = 0, kk algebraically closed). Sketch of proof.
    • Abstract intermezzo: smearing lemma: if M\otimes_A k(pp) is a d-dimensional vector space, then there is an f\in A\pp such that M_f is generated by at most d elements.
13. XIII lecture, 8.01.2020:
    • Graded abelian groups, rings and modules. Shifts.
    • Noetherian N-graded rings are finitely generated over noetherian A_0.
    • N-graded rings with A_0 a field and generated by A_1: Poincare characteristic and series, Hilbert polynomial. Poincare series is rational function. Graded Nagata trick and Noether normalization. The Poincare series of a graded algebra A has the form W/(1-T)^d, where d = dim(A) and 1-T does not divide W.
    • k^*-actions on graded objects, Proj(kk[x_1,...,x_n]), Proj_max(kk[x_1,...,x_n]) is in bijection with kkP^(n-1). Proofs will follow.
14. XIV lecture, 15.01.2020: very big picture of what we've done during the semester.
15. XV lecture, 22.01.2020:
    • Leftover: going-down lemma (without proof) and dim(A/pp) + dim(A_pp) = dim(A) for a prime ideal pp in a finitely generated k-algebra A that is a domain. Corollary: for every maximal mm in such an A we have dim(A) = dim(A_mm).
    • Completions. Definitions using Cauchy sequences and inverse limits. Complete local rings and Cohen's structure theorem (without the proof).

• broad picture understanding (when do we use this result/tool? what is the intuition?)
• knowledge (definitions, statements of results)
• technical skills (some ideas about proofs, some examples)

Pool one: prelims

1. Spectrum of a ring. Examples.
2. Localisation of rings and modules.
3. Spectrum of localization and quotient rings. Nilradicals.
4. Nakayama's lemma.
5. Tensor product of modules.
6. Derivations and cotangent module.
7. Fibers
8. Noetherian rings and modules. Hilbert basis theorem.
9. Integral and finite extensions. Normalization.

Pool two: main results

1. Krull dimension.
2. Noether's normalization.
3. Maximal ideals in finitely generated algebras. Nullstellensatz
4. Krull dimension for finitely generated algebras.
5. Algebraic sets in kk^n.
6. Discrete valuation rings and Dedekind domains.
7. Krull's principal ideal theorem and regular local rings.
8. Jacobian criterion
9. N-graded rings and their Noetherianity. N-graded finitely generated domains.

• Noether normalization.
• Noetherian, normal, one-dimensional local domain is a DVR.

Books and sources

• Some books:
  1. M.F. Atiyah, I.G. MacDonald. Wstep do algebry przemiennej. (main source)
  2. M. Reid. Undergraduate commutative algebra. (slightly easier and more geometric than previous.
  3. D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry more encyclopaedic than previous ones, look inside if curious.
  4. I. Kaplansky. Commutative Algebra. Great pure algebra book, older than others listed.
  5. mathoverflow contains many (counter)examples.

General evaluation rules