Melvin Hochster

Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay

Introduction 115 1. Some Applications of the Main Theorem 117 2. Counterexamples 124 3. Preliminaries and Basic Facts 127 4. Outline of the Proof 130 5. R&me of Cohomology 132 6. Pure Subrings 135 7. The Main Results in Characteristic p 141 8. Generic Freeness 146 9. Generic Uniform Convergence of Koszul Cohomology Along the Fibers 148 10. Resume of Invariant Theory 151 11. Reduction to the Graded Case 154 12. Proof of Proposition A, First Step: Descent 158 13. Proof of Proposition A, Second Step: The Key Argument in Characteristic p 162 14. Arithmetically Cohen-Macaulay Projective Schemes 165 15. Proof of Proposition B, First Step: Descent 168 16. Proof of Proposition B, Second Step: The Final Twist in Characteristic p 170 References 172

The purity of the Frobenius and local cohomology

rZ ring homomorphism (all rings are commutative, with identity, and homomorphisms preserve the identity) R --f S is called pure if for every R-module M, M ---f M RR S via m -+ m @ 1 is injective. Quite generally, if I is a homogeneous ideal of a graded ring R and rad(1) = rad((fo ,..., fi2)R), where f0 ,...,fn are forms of R, then the local cohomology modules H,i(R) can be expressed as direct limits of Koszul cohomology:

Comparison of symbolic and ordinary powers of ideals

All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I , then I (hn) ⊆ I n for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I , I (t) denotes the contraction of I t RW to R, where RW is the localization of R at the multiplicative system W . Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist. Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argument here is closely related to a proof given in [Hu, 5.14–16, p. 45] that regular local rings in characteristic p are UFDs,

Canonical elements in local cohomology modules and the direct summand conjecture

One of the objectives of this paper is to show that the usual homological consequences of the existence of big Cohen-Macaulay (henceforth, C-M) modules (e.g., the new intersection conjecture of Peskine-Szpiro and Roberts and the Evans-Griffith syzygy conjecture) follow from the direct summand conjecture when the residual characteristic of the local ring is positive. This gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive, and reduces the general case of all these conjectures to one rather down-to-earth conjecture. Of course, this places the direct summand conjecture in a position of central importance, so that it now merits an allout attack. Some partial results on this problem are given in Section 6. (The conjecture asserts that a regular Noetherian ring R is a direct summand (as an R-module) of every module-finite extension ring S 3 R.) The other main objective of this paper is to formulate and develop a theory of certain “canonical elements” in the local cohomology of special modules of syzygies. (Neither the modules of syzygies nor the induced maps between them are canonical, but the identification between the canonical elements is independent of the choices.) In particular, a canonical element qR E HZ (syz” K) is associated (see Section 3 for details) with each ndimensional local ring (R, m, K) (rings are commutative, associative, with identity; “local ring” means Noetherian ring with a unique maximal ideal). The conjecture that qR is nonzero for all local rings R turns out to be equivalent to the direct summand conjecture: for a given R, an infinite family of cases of the latter implies the former.

Localization of tight closure and modules of finite phantom projective dimension

The notion of tight closure for an ideal or submodule, both for Noetherian rings of positive prime characteristic p and for finitely generated algebras over a field of characteristic 0, was introduced in [HH4]. Expository accounts are given in [HH 1]-[HH3] and [Hu 1], The theory is further explored in [HH 6], [HH 8], [HH 9], and [HH 10], äs well äs in [Ab]. In particular, [HH8] contains a detailed study of the notion of phantom homology (homology is phantom when the cycles are in the tight closure of the boundaries in the module of chains), which leads to the theory of modules of finite phantom projective dimension developed in [Ab], an important tool here. The definition of finite phantom projective dimension and the parts of the theory that we need are reviewed briefly in § 5.

A Nullstellensatz with Nilpotents and Zariski's Main Lemma on Holomorphic Functions

The classical Nullstellensatz asserts that a reduced affine variety is known by its closed points; algebraically, a prime ideal in an affine ring is the intersection of the maximal ideals containing it. A leading special case of our theorem says that any affine scheme can be distinguished from its subschemes by its closed points with a bounded index of nilpotency; algebraically, an ideal I in an affine ring A may be written as I = (I (me + I), (*I n*ev

Tight closure and elements of small order in integral extensions

Throughout this paper ‘ring’ means commutative ring with identity and modules are unital. Given rings are usually assumed to be Noetherian, but there are notable exceptions. The phrase ‘characteristic p’ always means ‘positive prime characteristic p’, and the letters q, q’, etc. denote pe, p”, etc., where e, e’ E N, the nonnegative integers. The authors have recently introduced the notion of tight closure for a submodule N of a finitely generated module M over a Noetherian ring R of characteristic p and in certain equicharacteristic zero cases, including affine algebras over fields of characteristic 0. The theory started with the study of the notion of tight closure for an ideal I c R, i.e. with the case M = R, N = I, and this is still perhaps the most important case. The notion of tight closure has yielded new proofs, and, in many instances, unexpectedly strong improvements, of the local homological conjectures, of the existence of big Cohen-Macaulay modules, of the Cohen-Macaulay property for subrings which are direct summands of regular rings (where A is a direct summand of R means that A is a direct summand of R as an A-module) in the equicharacteristic case (in fact, tight closure techniques give the first proof of this fact in complete generality in the

The Zariski-Lipman Conjecture in the Graded Case

Let K be a field of characteristic 0, let R be a finitely generated reduced K-algebra, and let P be a prime ideal of R. The Zariski-Lipman conjecture asserts that if Der,(R, , Rp) (which may be identified with @er,(R, R))p) is R,-free, then R,, is regular. It is known that if Der,(R, , Rp) is R,-free, then Rp is a normal domain [SJ and in the case where either R is a hypersurface [7,8] or else R is a homogeneous complete intersection and P is the irrelevant ideal [6] (also, [4]) the conjecture has been verified. Our main objective here is to prove the conjecture in the case R = OF-, R, is graded by the nonnegative integers N, R, = K, and P = m, where m = @YE, Ri is the irrelevant maximal ideal. (We do not require that R be generated by its one-forms.) The paper concludes with a section containing several remarks about the inhomogeneous case, including a criterion for the freeness of the module of derivations of a two-dimensional local complete intersection which we feel ,may lead to a counterexample.

Finite Tor dimension and failure of coherence in absolute integral closures

It is shown both in characteristic p > 0 and in mixed characteristic p > 0 that if R is a perfect ring in the first case or RpR is perfect in the second case, then, under some additional conditions, the radical of a finitely generated ideal has finite Tor dimension, and bounds are obtained. Let R+ denote the integral closure of the domain R in an algebraic closure of its fraction field. The results are applied to show that R+ is not coherent when R is Noetherian of dimension at least 3, and, under additional restrictions, when the dimension is 2. Motivation for this question connected with tight closure theory is discussed.

The Zariski-Lipman conjecture for homogeneous complete intersections

A new short proof is given that if R is a homogeneous complete intersection over a field K of char 0 and Der (R, R) is R-free, then R is a polynomial ring. Let K be a field with char K = 0. The Zariski-Lipman conjecture as-