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Math 711: Lecture of December 7, 2005
We now use the notion of trace developed last time to prove:
Theorem. Let R be a Noetherian ring containing Q and let S be a module-finite extension of R such that pdRS < ∞. Then R is a direct summand of S.
Proof. The issue is local on R, and so we may assume that R is local. Each element s ∈ S gives an R-linear endomorphism φs of S, namely multiplication by s. The injection R ↪→ S shows that if we localize at the multiplicative system W of all nonzerodivisors in S, W −^1 S
is W −^1 R-free of rank ρ at least one. The map s 7 →
ρ
Tr(φs) gives an R-linear retraction
from S → R.
Remark. Of course the proof shows that the result holds whenever ρ is invertible in R: we do not need to assume that R contains Q.
We next discuss some variant notions of tight closure: we shall use one of these to prove a strong form of a result of Evans and Griffith on ranks of modules of syzygies over a regular local ring.
Given a non-empty family of nonzero ideals C in a Noetherian ring R of characteristic p > 0 with the property
(∗) if C, C′^ ∈ C then there exists C′′^ ∈ C such that C′′^ ⊆ C ∩ C′
we can define the tight closure with respect to C: an element u ∈ N ⊆ M is in the tight closure with respect to C of N in M if there exists an ideal C ∈ C such that Cuq^ ∈ N [q]^ for all q = pe^ 0. We can also define the small tight closure of N in M with respect to C: for this we require that for some C ∈ C, Cuq^ ∈ N [q]^ for all q (which includes q = 1). The property (∗) is needed so that the tight closure of N will be closed under addition.
If we take the family C to consist of all principal ideals generated by an element of R◦, we obtain the usual notion of tight closure.
If the family consists of only the unit ideal R, tight closure with respect to this family is Frobenius closure, while the small tight closure of N is the submodule N itself.
If R has a test element, tight closure with respect to the family consisting of the single ideal it generates gives ordinary tight closure, as does small tight closure with respect to the family consisting of the single ideal it generates.
We note that iterating one of these variant tight closure operations may give a larger result than performing it once. One can show that iterating the operation gives the same result if the family of ideals has the property that for all C, C′^ ∈ C, there exists C′′^ ∈ C such that C′′^ ∈ CC′.
We now want to show how one of these variant notions of tight closure can by used to prove the Evans-Griffith syzygy theorem. We want to make two remarks. First, it is immediate from the definition that u ∈ M is in the tight closure (respectively, small 1
2
tight closure) with respect to C of N in M if and only if the image of u in M/N is in the tight closure (respectively, small tight closure) of 0 in M/N with respect to C. The second remark we state as:
Lemma. If (R, m, K) is local, C is a non-empty family of nonzero ideals of R, and x is a minimal generator of a finitely generated module M , then x is not in the tight closure (nor in the small tight closure) of 0 in M with respect to C.
Proof. If u is in the tight closure of 0 in M we have that Cxq^ = 0 in F e(M ) for all q 0. We can map M K so that x 7 → 1. We get an induced surjection F e(M ) → R/m[q]. It follows that C ⊆ m[q]^ for all q 0, which implies that C = (0), a contradiction.
We shall need to make use of the notion of order ideal. Let x be an element of M , a finitely generated module over a Noetherian ring R. We define the order ideal OM (x) = O(x) to be {f (x) : f ∈ HomR(M, R)}. For finitely generated modules over a Noetherian ring R, the formation of the order ideal commutes with localization.
The map R → M sending 1 7 → x evidently splits if and only if OM (x) = R.
Also note that for any finitely generated free R-module G, any R-linear map M → G takes x into OM (x)G.
The Evans-Griffith syzygy theorem asserts that, a k th module of syzygies over a regular local ring, if not free, has rank at least k. They prove more general statements, in which the conditions on the ring are weakened but the module is assumed to have finite projective dimension. However, the key point in their proof is the following:
Theorem (Evans-Griffith). Let R be a local ring containing a field, let M be a k th module of syzygies of a finitely generated module of finite projective dimension, and suppose that MP is RP -free for every prime P of R except the maximal ideal, i.e., M is locally free on the punctured spectrum of R. Let x ∈ M be a minimal generator. Then O(x) is either the unit ideal or else has height at least k.
In fact, they show that this is true by using the fact that the improved new intersection theorem is true when R contains a field, which they deduce from the existence of big Cohen-Macaulay modules in the equal characteristic case. We shall eventually give their argument, but we first prove a better result in characteristic p, with depth replacing height and without the assumption that M is locally free on the punctured spectrum. We use a variant notion of tight closure in the argument.
Theorem. Let (R, m, K) be a local ring of prime characteristic p > 0 and let N be a finitely generated module of finite projective dimension over R. Let M be a finitely generated k th module of syzygies of N , and let x ∈ M be a minimal generator of M. Let I = OM (x). Then either I = R or else depthI R ≥ k.
Proof. If not, let y 1 ,... , yd be a maximal regular sequence in the proper ideal I, and let J = (y 1 ,... , yd)R. Then we can choose c ∈ R − J such that cI ⊆ J. Let c′^ denote the image of c in R′^ = R/J. Let G• be a resolution of N by finitely generated free modules over R such that Gk → Gk− 1 factors Gk M ↪→ Gk− 1 , which we know exists because M is k th module of syzygies of N over R. Let B denote the image of Gk+1 in Gk. Let
