# COURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA

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1 COURSE SUMMARY FOR MATH 508, WINTER QUARTER 2017: ADVANCED COMMUTATIVE ALGEBRA JAROD ALPER WEEK 1, JAN 4, 6: DIMENSION Lecture 1: Introduction to dimension. Define Krull dimension of a ring A. Discuss dimension 0 rings. Recall Artinian rings and various equivalences. Prove that a PID has dimension 1. Prove that if I R is a nilpotent ideal, then dim R/I = dim R. Lecture 2: Conservation of dimension under integral extensions. Prove that if R S is an integral ring extension, then dim R = dim S. Define the codimension (or height) of a prime p A, denoted as codim p, as the supremum of the lengths k of strictly descending chains p = p 0 p 1 p k of prime ideals. Note that codim p = dim A p. Prove: if φ: R S is an integral ring homomorphism, then dim I = dim φ 1 (I). Discuss codimension 0 primes (i.e. minimal primes). Prove: if R is Noetherian and f R is a non-unit, then any prime p (f) has codim(p) = 0. WEEK 2, JAN 9, 11 (JAN 13 CANCELLED): KRULL S HAUPTIDEALSATZ AND CONSEQUENCES Lecture 3: Krull s Hauptidealsatz. State and prove Krull s Principal Ideal Theorem (a.k.a. Krull s Hauptidealsatz): if A is a Noetherian ring and f A is not a unit, then height(f) 1; that is, for every prime ideal p containing f, height p 1. State and prove the following generalization of Krull s Principal Ideal Theorem: if A is a Noetherian ring and I = (x 1,..., x n ) A is a proper ideal. Then height I n; that is height p n for every prime ideal p containing I (or equivalently, for every prime ideal which is minimal among prime ideals containing I). Prove corollary: dim k[x 1,..., x n ] = n. 1

2 2 JAROD ALPER Macaulay2 Tutorial. (Evening of Jan 9) Lecture 4: System of parameters. Prove the converse theorem to Krull s principal ideal theorem: if A is a Noetherian ring and I A is a proper ideal of height n. Then there exist x 1,..., x n I such that height(x 1,..., x i ) = i for i = 1,..., n. Reinterpret dimension: if (R, m) is a Noetherian local ring, them dim R is the smallest number n such that there exists x 1,..., x m m with R/(x 1,..., x m ) Artinian. Such a sequence x 1,..., x m m is called a system of parameters for R. Prove corollary: if (R, m) (S, n) is a local homomorphism of local Noetherian rings, then dim S dim R + dim S/mS. Prove corollary: if R is a Noetherian ring, then dim R[x] = dim R + 1. WEEK 3, JAN 18, 20 (JAN 16 MLK HOLIDAY): FLATNESS Lecture 5: Basics on flatness. Review of Tor. Key properties: short exact sequences induce long exact sequences of Tor groups, Tor i (P, M) = 0 for P projective and i > 0, compatibility with localization, Tor i (M, N) = Tor i (N, M) and thus can be computed as a derived functor in either the first or second term. Examples of flat and non-flat modules Prove Going Down Theorem for flatness Prove: if (R, m) (S, n) is a flat local homomorphism of local Noetherian rings, then dim S = dim R + dim S/mS. Lecture 6: Homological characterization of flatness: Prove: Let R be a ring. An R-module M is flat if and only if Tor R 1 (R/I, M) for all finitely generated ideals I R. Examples: flatness over the dual numbers, flatness over PIDs. Equational Criterion for Flatness: An R-module M is flat if and only if the following condition is satisfies: For every relation 0 = i n im i with m i M and n i R, there exist elements m j M and elements a ij R such that a ij m j = m i for all i and a ij n i = 0 for all j. j WEEK 4, JAN 23, 25, 27: ARTIN REES LEMMA, KRULL S INTERSECTION THEOREM, LOCAL CRITERION OF FLATNESS Lecture 7: flatness projective. Reinterpret equational criterion for flatness using commutative diagrams. i

3 SUMMARY OF MATH Prove: if M is finitely presented R-module, then M is flat projective. If in addition R is a local, then flat free projective. State and motivate simple version of Local Criterion for Flatness. Lecture 8: Artin Rees Lemma and Local Criterion for Flatness. Give motivation of the Artin Rees Lemma. Prove Artin Rees Lemma: Let R be a Noetherian ring and I R an ideal. Let M M be an inclusion of finitely generated R- modules. If M = M 0 M 1 M 2 is an I-stable filtration, so is M M M 1 M M 2. Prove Krull s Intersection Theorem: Let R be a Noetherian ring, I R an ideal and M a finitely generated R-module. Then there exists x I such that (1 x) k I k M = 0. In particular, if (R, m) is local, then m k M = 0 and m k = 0. Or if R is a Noetherian domain and I is any ideal, then I k = 0. State general version of Local Criterion for Flatness: if (R, m) (S, n) is a local homomorphism of local Noetherian rings and M is a finitely generated S-module, then M is flat as an S-module Tor R 1 (R/m, M) = 0. Lecture 9: Fibral Flatness Theorem. Finish proof of Local Criterion for Flatness. Prove the Fibral Flatness Theorem: Consider a local homomorphisms (R, m) (S, n) (S, n ) of local Noetherian rings. Let M be a finitely generated S -module which is flat over R. Then M is flat over S if and only if M/mM is flat over S/mS. Discuss special case of the Fibral Flatness Theorem when R = k[x] (x). WEEK 5, JAN 30, FEB 1, 3: GRADED MODULES AND COMPLETIONS Lecture 10: Graded modules and flatness. Summary of flatness results. State Openness of Flatness and Grothendiecke s generic freeness (without proof). Graded modules and Hilbert functions. Prove: Let R = d 0 R d be a graded ring which is finitely generated as an R 0 -algebra by elements of degree 1. Assume R 0 is a local Noetherian domain. Let M be a finitely generated R-modules. Then M is flat/r 0 if and only if for p Spec R 0, the Hilbert function H M R0 k(p) := dim k(p) M d R0 k(p) is independent of p.

4 4 JAROD ALPER Lecture 11: Completions. Definition of the completion of a ring (and module) with respect to an ideal. Arithmetic and geometric examples. Show that if R is Noetherian and I R is an ideal, then M R ˆR ˆM is an isomorphism for all finitely generated modules M. Conclude that R ˆR is flat. Lecture 12: Completions continued. Show that if R is Noetherian, then so is the completion ˆR of R along an ideal I. Conclude that R Noetherian = R[[x]] Noetherian. Mention Hensel s Lemma. Mention Cohen s Structure Theorem. WEEK 6, FEB 8, 10 (FEB 6 SNOW DAY): REGULAR SEQUENCES AND KOSZUL COMPLEXES Lecture 13: regular sequences. Introduce regular sequences: we say x 1,..., x n R that is a M- regular sequence if x i is a non-zero divisor on M/(x 1,..., x i 1 )M for i = 1,..., n and that M (x 1,..., x n )M. Give examples. Prove: Let R be a ring and x 1,..., x n be a regular sequence. Set I = (x 1,..., x n ). Show that the natural homomorphism R/I[y 1,..., y n ] Gr I R, y i x i I/I 2, is an isomorphism. In particular, I/I 2 is a free R/I-module of rank n. State more general version (which has the same proof) when M is an R-module and x 1,..., x n is a M-regular sequence. Lecture 14: Koszul complex. Finish proof of proposition from last class. Introduce alternating products. Give concrete definition of the Koszul complex: if R is a ring, M is an R-module and x = (x 1,..., x n ) R n, then K(x; M) = K(x 1,..., x n ; M) is the chain complex of R-modules where K(x; M) k = k (R n ) and the differential k (R n ) k 1 (R n ) is defined by e i1 e ik k j=1 ( 1)j x ij e i1 ê ij e ik. Write down examples in the case n = 1, 2, 3. WEEK 7, FEB 13, 15, 17: KOSZUL HOMOLOGY, DEPTH AND REGULARITY Lecture 15: Koszul complexes and regular sequences. Reintrepret the Koszul complex as a tensor product of complexes.

5 SUMMARY OF MATH Define Koszul homology: if R is a ring, M is an R-module and x = (x 1,..., x n ) R n, then H i (x; M) := H i (K(x; M)). Prove: Let R be a ring, M be an R-module and x = (x 1,..., x n ) R n. If x is an M-regular sequence, then H i (x; M) = 0 for i 1. Show that the converse is true if (R, m) is a local Noetherian ring, M is a finitely generated R-module and x 1,..., x n m. Lecture 16: depth. Finish proof of converse from previous lecture. State the theorem: Let R be a Noetherian ring and M be a finitely generated R-module. Let x = (x 1,..., x n ) R n and set I = (x 1,..., x n ) R. Assume IM M. Let d be the smallest integer such that H n d (x; M) 0. Then any maximal M-regular sequence in I has length d. Define the depth of I on M, denoted by depth(i, M), as this smallest integer d. Give examples. Begin proof of theorem. Lecture 17: depth and regular local rings. Finish proof characterizing depth. State (but do not prove): Let R be a Noetherian ring, I R be an ideal and M be a finitely generated graded R-module. Assume I + Ann(M) R. Then depth(i, M) is the smallest i such that Ext i R(R/I, M) 0. Define a regular local ring (R, m) as a Noetherian local ring such that dim R = dim R/m m/m 2. Give a few examples: e.g., say when k[x 1,..., x n ] (x1,...,x n)/(f) is regular. Prove that a regular local ring is a domain. WEEK 8, FEB 22, 24 (FEB 20 PRESIDENT S DAY): FREE RESOLUTIONS AND PROJECTIVE DIMENSION Lecture 17: minimal free resolutions and projective dimension. Show that if (R, m) is a regular local ring and x 1,..., x k are elements of m which are linearly independent in m/m 2, then x 1,..., x k is a regular sequence and R/(x 1,..., x k ) is a regular local ring. Such a sequence whose length is equal to dim R is called a egular system of parameters. Introduce the projective dimension of a module M, denoted by pd M, as the smallest length of a projective resolution of M. Introduce the global dimension of a ring R, dentoed by gl dim M, as the supremum of pd M over finitely generated R-modules M.

6 6 JAROD ALPER Define a minimal free resolution of an R-module M as a free resolution F d k k Fk 1 F 0 M such that im(d k ) mf k 1. Show: if (R, m) is a local Noetherian ring and M is a finitely generated R-module, then pd M is equal to length of any minimal free resolution and is also characterized by the smallest i such that Tor R i+1(r/m, M) = 0. Conclude that gl dim R = pd R/m Lecture 18: Auslander Buchsbaum Theorem. Recall notions and results from previous lecture. Show that if (R, m) is a regular local ring, then dim R = gl dim R/m. Discuss examples showing that this is not true if R is not regular. Discuss (without proof) graded analogues: minimal graded free resolutions, graded Betti numbers and namely Hilbert s Syzygy Theorem (any finitely generated graded module over the polynomial ring k[x 1,..., x n ] has a finite free graded resolutions of length n. Recall the notion of depth and prove that if R is a Noetherian ring and I R is an ideal, then depth(i, R) codim I.. Prove the Auslander-Buchsbaum Theorem: Let (R, m) be a Noetherian local ring and M 0 be a finitely generated R-module with pd M <. Then pd M = depth(m, R) depth(m, M). WEEK 9, FEB 27, MAR 1 (MAR 3 CANCELLED): THE AUSLANDER BUCHSBAUM SERRE THEOREM AND COHEN MACAULAY RINGS Lecture 19: Homological characterization of regular rings. Prove the Auslander Buchsbaum Serre Theorem: If (R, m) is a Noetherian local ring, the following are equivalent: (i) R is regular. (ii) gl dim R <. (iii) pd R/m <. Lecture 20: Cohen-Macaulay rings. Prove the following corollaries of the Auslander Buchsbaum Serre Theorem: If R is a regular local ring, then so is R p for every prime ideal p. k[x 1,..., x n ] is a regular ring (i.e., all localizations at prime ideals are regular local rings). Define a Noetherian local ring (R, m) to be Cohen Macaulay if depth(m, R) = dim R.

7 SUMMARY OF MATH Show that the following examples of Cohen Macaulay rings: (1) regular local rings, (2) local Artinian rings, and (3) local Noetherian dimension 1 reduced rings. Prove: If (R, m) is a Cohen Macaulay local ring and p is an associated prime, then p is a minimal prime and dim R = dim R/p. (In other words, Cohen Macaulay rings can have no embedded primes (i.e. associated but not minimal primes) and is equidimensional.) Give some examples of rings that are not Cohen Macaulay: k[x, y]/(x 2, xy), k[x, y, z]/(xz, yz),... WEEK 10, MAR 6, 8, 10: COHEN MACAULAY, NORMAL, COMPLETE INTERSECTIONS AND GORENSTEIN RINGS Lecture 21: Properties of Cohen-Macaulay rings and Miracle Flatness. Prove: If (R, m) is a Cohen Macaulay local ring, then for any ideal I R, we have depth(i, R) = dim R dim R/I = codim I. (In particular, the defining property of being Cohen Macaulay holds for all ideals. Also, in the homework, we will see that Cohen Macaulay rings satisfy a stronger dimension condition known as catenary.) Prove: Let (R, m) be a Cohen Macaulay local ring. Then x 1,..., x n m is a regular sequence if and only if dim R/(x 1,..., x n ) = dim R n. In other words, if (R, m) is a Cohen Macaulay local ring, then any system of parameters is a regular sequence. Prove Miracle Flatness: Let (R, m) (S, n) be a local homomorphism of Noetherian local rings. Suppose that R is regular and S is Cohen Macaulay. Then R S is flat if and only if dim S = dim R + dim S/mS. Lecture 22: Complete Intersections and Normal rings. Define a Noetherian local ring (R, m) to be a complete intersection if the completion ˆR is the quotient of a regular local ring by a regular sequence. Observe that any regular local ring modulo a regular sequence is a complete intersection. Show that any complete intersection local ring is Cohen Macaulay. Give an example of a Cohen Macaulay ring (e.g. k[x, y]/(x, y) 2 ) which is not a complete intersection. Given a Noetherian local ring (R, m) with residue field k = R/m and minimal generators x 1,..., x n of m, define the invariants ɛ i (R) = dim k H i (x 1,..., x n ) (the Koszul homology). The number of minimal generators is called the embedding dimension of R is denoted emb dim(r). State: R is a complete intersection if and only if dim R = emb dim R ɛ 1 (R). Give several examples of both when this holds and doesn t.

8 8 JAROD ALPER Recall that a domain R is called normal if it is integrally closed in its fraction field. For a Noetherian ring R, introduce Serre s properties: (R i ) for all p Spec R with codim(p) i, R p is regular. (S i ) for all p Spec R, depth R p min(codim(p), i). Note that: R is regular if and only if (R i ) holds for all i and R is Cohen Macaulay if and only if (S i ) holds for all i. Reinterpret the Conditions (S 0 ), (S 1 ) and (S 2 ). State that R is reduced if and only if (R 0 ) and (S 1 ) hold. State Serre s Normality Criterion: Let R be a Noetherian ring. Then R is normal if and only if (R 1 ) and (S 2 ) hold. (Prove the implication.) Mention Algebraic Hartog s: If R is a normal Noetherian domain, then R = codim(p)=1 R p. Lecture 23: Gorenstein rings. Recall the notion of an injective module and an injective resolution. If R is a ring, define the injective dimension of an R-module, denoted by inj dim R M, as the smallest length of an injective resolution of M. State (and explain some of the implications in) the following lemma: If R is a ring and M is an R-module, then the following are equivalent: (i) inj dim R M n. (ii) Ext n+1 R (N, M) = 0 for all R-modules N. (iii) Ext n+1 R (R/I, M) = 0 for all ideals I R. If, in addition, (R, m) is local and M is finitely generated, then the above is also equivalent to: (iv) Ext n+1 R (R/p, M) = 0 for all p Spec R. If, in addition, (R, m) is a Noetherian local ring and M is finitely generated, then the above is also equivalent to: (iv) Ext n+1 R (R/m, M) = 0. Conclude: (R, m) is a Noetherian local and M is a finitely generated R-module, then inj dim R M is the largest i such that Ext i R(R/m, M) 0. State: If (R, m) is a Noetherian local and M is a finitely generated R-module with inj dim R M <, then dim M inj dim R M = depth(m, R). Compare the above characterizations and properties of injective dimension with what we ve seen for projective dimension. Define a Noetherian local ring (R, m) to be Gorenstein if inj dim R R <. Give the following equivalences (explaining some of the implications): Let (R, m) is a Noetherian local ring of dimension n with residue field k = R/m, then the following are equivalent

9 SUMMARY OF MATH (i) R is Gorenstein. (ii) inj dim R R = n. (iii) Ext i R(k, M) = 0 for some i > n. (iv) { 0 i n Ext i R(k, M) = k i = n. (v) R is Cohen Macaulay and Ext n R(k, R) = k. (vi) There exists a regular sequence x 1,..., x n m such that R/(x 1,..., x n ) is Gorenstein and dimension 0. Prove: Let (R, m) be a Noetherian local ring. Then regular = complete intersection = Gorenstein = Cohen Macaulay Give examples showing that each implication is strict.

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