Cohen-Macaulay Rings: Proof of Theorem (September 19, 2005) - Prof. Melvin Hochster | Study notes Algebra

Math 711: Lecture of September 19, 2005

Our next objective is to prove the Cohen-Macaulayness assertions in the statement of

the second Theorem of the Lecture of September 12. The argument is entirely similar to

what we did earlier in studying the ideal generated by the 2 ×2 minors of a matrix of

indeterminates.

We use reverse induction, assuming the result that larger ideals of the form Iσ+Jsk(X)

are Cohen-Macaulay.

Suppose that a specific prime of the form Iσ+Jsk(X) is given. Call the ideal P. To

show that K[X]/P is Cohen-Macaulay, it suffices to show that the depth of K[X]/P on

the ideal mgenerated by all the xi,j in K[X] is d= dim (K[X]/P ): we review the relevant

facts about the Cohen-Macaulay property below. Let x=x1,sk+1. Since we already

know that K[X]/P is a domain, we have that xis a nonzerodivisor, and so K[X]/P

is Cohen-Macaulay if and only if K[X]/(P+xK[x]) is, and this may be described as

K[X]/(Iσ(X) + Jsk+1(X)).

There are two cases. If sk+1 = sk+1 , then I+xK[X] is a larger prime ideal of our family,

and so killing it gives a Cohen-Macaulay ring by the induction hypothesis. If sk+ 1 < sk+1

then I+ (x) is radical. By the lemma on killing minors, each of the variables x1,b for

sk+ 1 < b < sk+1 kills Ik(X|sk+1 ). Let σ0be the result of changing skin σto sk+ 1, while

leaving all other entries fixed. Let Q1=Iσ0+Jsk+1(X) and Q2=Iσ(X) + Jsk+1(X). Both

of these ideals are prime, and we know that they have Cohen-Macaulay quotients by the

induction hypothesis. This is also true for Q3=Q1+Q2=Iσ0(X) + Jsk+1 (X).

Note that V(P+ (x)) = V(Q1)∪V(Q2) by the lemma on killing minors: since all of

the ideals are radical, we have that P=Q1∩Q2.

Moreover, K[X]/Q1has dimension d−1: among the numbers used in calculating the

dimension, skhas increased by one while all others, including k, have not changed. Sim-

ilarly, K[X]/Q2has dimension d−1: here, only khas changed, increasing by 1. Finally,

K[X]/Q3has dimension d−2, since in this case khas increased by 1 and skhas increased

by one. Since these are Cohen-Macaulay, in the short exact sequence

0→K[X]/(P+ (x)) →K[X/Q1⊕K[X]/Q2→K[X]/Q3→0

the depths of the middle and right hand terms on mare d−1 and d−2 respectively, and

so the depth of K[X]/(P+ (x)) is d−1, as required. 

We next review the basic facts about Cohen-Macaulay rings.

x1, . . . , xn∈Ris called a possibly improper regular sequence on the R-module Mif

x1is not a zerodivisor on Mand for every i, 1 < i ≤n,xiis not a zerodivisor on

M/(x1, . . . , xi−1)M. It is called a regular sequence if, moreover, (x1, . . . , xn)M6=M.

Let R→Sbe a homomorphim of Noetherian rings, Ian ideal of R, and Ma finitely

generated S-module. Then Mis also an R-module by restriction of scalars. If IM 6=Mwe

define the depth of Mon Ior depthIMto be the supremum of lengths of regular sequences

in Ion M. By a theorem, this supremum is finite, every regular sequence can be extended

to a maximal one, and all maximal regular sequences have the same length, which is the

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Math 711: Lecture of September 19, 2005

Our next objective is to prove the Cohen-Macaulayness assertions in the statement of the second Theorem of the Lecture of September 12. The argument is entirely similar to what we did earlier in studying the ideal generated by the 2 × 2 minors of a matrix of indeterminates. We use reverse induction, assuming the result that larger ideals of the form Iσ + Jsk (X) are Cohen-Macaulay. Suppose that a specific prime of the form Iσ + Jsk (X) is given. Call the ideal P. To show that K[X]/P is Cohen-Macaulay, it suffices to show that the depth of K[X]/P on the ideal m generated by all the xi,j in K[X] is d = dim (K[X]/P ): we review the relevant facts about the Cohen-Macaulay property below. Let x = x 1 ,sk +1. Since we already know that K[X]/P is a domain, we have that x is a nonzerodivisor, and so K[X]/P is Cohen-Macaulay if and only if K[X]/(P + xK[x]) is, and this may be described as K[X]/(Iσ (X) + Jsk +1(X)). There are two cases. If sk +1 = sk+1, then I +xK[X] is a larger prime ideal of our family, and so killing it gives a Cohen-Macaulay ring by the induction hypothesis. If sk + 1 < sk+ then I + (x) is radical. By the lemma on killing minors, each of the variables x 1 ,b for sk + 1 < b < sk+1 kills Ik(X|sk +1). Let σ′^ be the result of changing sk in σ to sk + 1, while leaving all other entries fixed. Let Q 1 = Iσ′ + Jsk +1(X) and Q 2 = Iσ (X) + Jsk+1 (X). Both of these ideals are prime, and we know that they have Cohen-Macaulay quotients by the induction hypothesis. This is also true for Q 3 = Q 1 + Q 2 = Iσ′^ (X) + Jsk+1 (X). Note that V (P + (x)) = V (Q 1 ) ∪ V (Q 2 ) by the lemma on killing minors: since all of the ideals are radical, we have that P = Q 1 ∩ Q 2. Moreover, K[X]/Q 1 has dimension d − 1: among the numbers used in calculating the dimension, sk has increased by one while all others, including k, have not changed. Sim- ilarly, K[X]/Q 2 has dimension d − 1: here, only k has changed, increasing by 1. Finally, K[X]/Q 3 has dimension d − 2, since in this case k has increased by 1 and sk has increased by one. Since these are Cohen-Macaulay, in the short exact sequence

0 → K[X]/(P + (x)) → K[X/Q 1 ⊕ K[X]/Q 2 → K[X]/Q 3 → 0

the depths of the middle and right hand terms on m are d − 1 and d − 2 respectively, and so the depth of K[X]/(P + (x)) is d − 1, as required.

We next review the basic facts about Cohen-Macaulay rings. x 1 ,... , xn ∈ R is called a possibly improper regular sequence on the R-module M if x 1 is not a zerodivisor on M and for every i, 1 < i ≤ n, xi is not a zerodivisor on M/(x 1 ,... , xi− 1 )M. It is called a regular sequence if, moreover, (x 1 ,... , xn)M 6 = M. Let R → S be a homomorphim of Noetherian rings, I an ideal of R, and M a finitely generated S-module. Then M is also an R-module by restriction of scalars. If IM 6 = M we define the depth of M on I or depthI M to be the supremum of lengths of regular sequences in I on M. By a theorem, this supremum is finite, every regular sequence can be extended to a maximal one, and all maximal regular sequences have the same length, which is the 1

depth. If IM = M the depth is defined to be +∞. It turns out the depthI M = depthIS M. The most important case is where S = R, but it is often useful to have the greater generality available. REFS By a theorem, if d = depthI M and N is any R-module with annihilator I, then ExtiR(N, M ) = 0 for i < d while ExtdR(N, M ) 6 = 0. In particular, one may choose N = R/I, and the long exact sequence for Ext may then be used to prove facts about the behavior of depth in short exact sequences. For example, if 0 → M ′^ → M → M ′′^ → 0 is exact, and depthI M ′′^ < depthI M , then depthI M ′^ = depthI M ′′^ + 1. We used this last assertion earlier, when the depth of the middle term was d − 1 and the depth of the right hand term was d − 2. A local ring (R, m, K) is called Cohen-Macaulay if some (equivalently, every) system of parameters is a regular sequence in the ring. Regular local rings are Cohen-Macaulay and so are their quotients by an ideal generated by part of a system of parameters: the latter are called local complete intersections. If R is a Cohen-Macaulay local ring, then each of its localizations at a prime is again Cohen-Macaulay. REFS A Noetherian ring R is called Cohen-Macaulay if all of its localizations at prime (equiv- alently, at maximal) ideals are Cohen-Macaulay local rings. The quotient of Cohen- Macaulay ring by a proper ideal generated by a regular sequence is again Cohen-Macaulay. All Noetherian rings of dimension 0 are Cohen-Macaulay, and all reduced Noetherian rings of dimension 1 are Cohen-Macaulay as well. Of course, this includes the one-dimensional domains. By a theorem, all normal Noetherian domains of dimension two are Cohen- Macaulay. REFS If a local ring R is a module-finite extension of a regular local ring A, then R is Cohen- Macaulay if and only if it is free as an A-module. REF We want to focus on the graded case. Let R be an N-graded finitely generated algebra over a field K with R 0 = K. Let m denote the unique homogeneous maximal ideal: it

is spanned by all forms of positive degree, and may also be described as

⊕^ ∞

Rn. Let

dim (R) = d. By a theorem, there is always a sequence f 1 ,... , fd of forms in m such that Rad

(f 1 ,... , fd)

= m. Such a sequence is called a homogeneous system of parameters. Note that if K is infinite and R is generated by R 1 over K, then these may be chosen in R 1. In general it is necessary to use forms of degree larger than 1. If f 1 ,... , fd is a homogeneous system of parameters for R, R is module-finite over its subring K[f 1 ,... , fd], and this subring is a polynomial ring, i.e., f 1 ,... , fd are algebraically independent over K. The image of a homogeneous system of paramaters for R in Rm is a system of parameters for Rm.

Theorem. R be an N-graded finitely generated K-algebra as above, where K is a field, with R 0 = K and let m be the homogeneous maximal ideal of R. Suppose that dim (R) = d. The following conditions are equivalent: (1) R is Cohen-Macaulay. (2) Rm is Cohen-Macaulay. (3) Some (equivalently, every) homogeneous system of parameters for R is a regular se- quence in R.

Cohen-Macaulay Rings: Proof of Theorem (September 19, 2005) - Prof. Melvin Hochster, Study notes of Algebra

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