This article establishes a concrete relation between order ideals of minimal generators and annihilator ideals. For a regular local ring R and ideal I the authors construct an R-module M with minimal generator having I as order ideal. Further, it is shown that most variability in idealtheoretic behavior of such order ideals is exhibited by modules of projective dimension one. The authors "introduce" the concept of *-orthogonality and use their syzygy theorem to show constraints on the size and height of a *-orthogonal set in a given finitely generated non-free module. The paper contains an application of the theory of order ideals to the binomial behavior of syzygy rank.

In this article we examine Koh's theorem [10] on splitting and extend the result to a new class of polynomials in mixed characteristic for which the degree can be large.

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring $R$ versus a hypersurface ring $R/(x^n)$. Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic $p$, with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by $p^s$, $s>0$, that have finite projective dimension over a hypersurface ring.

The new intersection theorem is used to derive a criteria for flat descent in the setting of integral ring extensions. Applications, such as “purity of branch locus” for extensions of normal domains, are noted.

Introduction. In this article we are going to concentrate on the canonical element conjecture due to M. Hochster as well as several of its ramifications. In [17] Hochster introduced a number of equivalent forms of this conjecture and proved it in the equicharacteristic case. One of the earliest forms, the direct summand conjecture, was proved by Hochster [15] a decade earlier under the same hypothesis (see also [16]). In 1980 Evans and Griffith [10] gave an affirmative answer to the syzygy problem for equicharacteristic local rings. In the course of their proof, they implicitly established a result for finite free complexes [10] that Hochster explicitly isolated in his article [17]. He referred to the new result as the “improved new intersection theorem” (henceforth INIT) since it is clear that INIT implies the new intersection theorem ([17]). Of course INIT remains a “conjecture” in the case of mixed characteristic. In the same article [17] Hochster pointed out that INIT was a consequence of the canonical element conjecture and later the first author [3] showed that the two conjectures are equivalent. Over the years several special cases of the canonical element conjecture have been proved and new equivalent forms have been introduced ([2], [3], [5], [6], [7], [8], [14]). The four main equivalent versions of this conjecture, i.e., the direct summand conjecture, the monomial conjecture, the canonical element conjecture and the

Let A and B denote local rings such that A=B/tB, where t is a regular nonunit, and let b denote an ideal in B such that the A-ideal a=b/(t) has codimension ⩾2. Let F be a reflexive OX-module, where X=SpecA-V(a). Under suitable conditions on A and B and assuming that ExtX2(F,F)=0 and ExtX1(F,OX)=0, it is shown in this article that the dual sheaf Fv can be extended to a reflexive coherent OY-module, where Y=SpecB-V(b). The infinitesimal procedure that leads to this sheaf extension makes use of the injective theory of sheaves. Applications to homomorphisms of divisor class groups come about as a consequence of this result, and a strong connection with Grothendieck's theorem on parafactoriality is drawn.

The Baer splitting problem from the 1930s is revisited, after which, using current knowledge about maximal Cohen-Macaulay modules, the structure of Baer modules over regular integral domains of higher Krull dimension is explored. In particular, the countably generated ones in the local case are shown to be free.

The injectivity of the restriction homomorphism on divisor class groups to hypersurfaces has been studied by Grothendieck, Danilov, Lipman, and Griffith & Weston, among others. In particular, when A is a Noetherian normal domain of equicharacteristic zero and A/fA satisfies R1, Spiroff established a map Cl(A)→Cl((A/fA)′), where (A/fA)′ represents the integral closure of A/fA, and gave some conditions for injectivity. In this paper, the authors continue in the same vein, but in the case of characteristic p>0. In addition, when the hypersurface A/fA is normal, they provide further enlightenment about the kernel of Cl(A)→Cl(A/fA). Finally, using the second author's previous results, they exhibit a new class of examples for which the kernel is non-trivial.

In a large part, the goal of this article is to expose the common ground between the authors’ recent articles [11] and [15] and to explore certain byproducts which emerge for module finite ring extensions A=B of normal domains. These by-products come in two flavors; one being a characterization of purity of branch locus for A=B (see Section 1), and the other being a partial description of the image of the map of divisor class groups ClðBÞ ! ClðAÞ (see Section 3). The framework in which our discussion takes place is derived from the first author’s article [11, Section 2] and can be described as follows. Let B denote an excellent local normal domain and let A=B represent a module finite ring extension in which A is also a normal domain. Let X 1⁄4 ðX1; . . . ;XnÞ be a set of n polynomial variables over B, with n chosen large enough to provide a surjection from L 1⁄4 B1⁄2X1; . . . ;Xn onto A. In the case

About twenty years ago we gave the original proof of the so-called “syzygy theorem” [6], at least in the case of a local integral domain containing a field. In the intervening years several different styles of proofs and generalizations have appeared (e.g., see our monograph [7, Chapter 3]). Perhaps the most notable of these were the characteristic p proof of Hochster and Huneke [12] and the generalization by Bruns [2] in which the minimal free complex is allowed some positive homology (see [3, 9.5.5]). In addition Dutta [5] and Hochster [11] combined to show that the crucial argument in [6] which rested upon the existence of maximal Cohen-Macaulay modules could be reduced to an application of the weaker canonical element theorem which holds in the equicharacteristic setting (see also Ogoma [13]). Our original proof [6] relied heavily on the fact we were able to show order ideals of minimal generators for kth syzygies of finite projective dimension must have grade at least k. We have long known that a single minimal generator with this property would suffice. Until now we remained unaware of any context in which one could actually apply this (no doubt well-known) fact. However, a recent examination of a graded version of the syzygy theorem in mixed characteristic produced just such a situation. To be specific, we are able to show (see Theorem 8 and Corollary 9), under suitable conditions, a nonfree graded kth syzygy of finite projective dimension over a graded ring R = R0 ⊕R1 ⊕R2 ⊕ · · · must have rank at least k, where the ring R0 is a discrete valuation ring. Other than the observation described above, our techniques are simply a straightforward blend of those from our original manuscript and standard facts about graded rings and modules. In order to keep the exposition somewhat self contained, we supply a few elementary observations which can be found in some form in our articles [6], [7], [8] or Bruns-Herzog [3, Chapter 9]. These observations or facts about syzygies have been tailored to the context of graded modules over R = R0 ⊕ R1 ⊕ R2 ⊕ · · · as described above.

. The main result states: if A=B is a module (cid:12)nite extension of excellent local normal domains which is unrami(cid:12)ed in codimension two and if S= { S ’ ^ B represents a deformation of the completion of B , then there is a corresponding S -algebra deformation T= { T ’ ^ A such that the ring homomorphism S , ! T represents a deformation of ^ B , ! ^ A . The main application is to the ascent of the arithmetic Cohen-Macaulay property for an (cid:19)etale map f : X ! Y of smooth projective varieties over an algebraically closed (cid:12)eld. (cid:3)

For an extension R ↪ A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A → Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the 'splitting' (or 'branching') and the 'ramification' that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a 'normal point' of Spec R, that is, over p ∈ Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be 'unibranched', i.e., the Henselization of R/p is a (normal) domain].

In this article it is noted that, in equicharacteristic zero, the existence of small Cohen-Macaulay modules may be reduced to whether countably generatd infinite syzygies over cyclic hypersurface rings have finite type direct summands (Theorem 1.9). In the special case of simple singularities it is then shown (Corollary 5.2) that countably generated infinite syzygies decompose into direct sums of finitely generated modules.

For a module M M over a local Cohen-Macaulay ring R R we develop a (finite) sequence of presentations of M M which facilitates the study of invariants arising from the cohomology modules of M M . As an application we use this data, in case R R is regular and M M represents a vector bundle on the punctured spectrum of R R with a vanishing cohomology module, to obtain bounds on how far M M can be lifted as a vector bundle.

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Let R R be a local noetherian domain with algebraically closed residue field and let M M be a finitely generated module of rank r r which is not free. Then there is some minimal generator x x of M M such that the ideal of images of x x under maps of M M to R R has height at most r r .