Generalized divisors on Gorenstein schemes

We develop a theory of generalized divisors on a Gorenstein scheme whereby any closed subscheme of pure codimension one without embedded points can be regarded as an effective divisor. Most of the usual theory of linear equivalence, associated sheaf, etc., carries over to this more general setting. The definition uses reflexive sheaves, so we first review the theory of reflexive modules. As an application, we give new definitions of liaison and biliaison for subschemes of ⁿ , which simplify the foundations of the theory of liaison. We also compute explicitly the set of generalized divisor classes on some reducible and singular schemes.     

Classifying subcategories of modules over a commutative noetherian ring

Let R be a quotient ring of a commutative coherent regular ringby a finitely generated ideal. Hovey gave a bijection betweenthe set of coherent subcategories of the category of finitelypresented R-modules and the set of thick subcategories of thederived category of perfect R-complexes. Using this bijection,he proved that every coherent subcategory of finitely presentedR-modules is a Serre subcategory. In this paper, it is provedthat this holds whenever R is a commutative noetherian ring.This paper also yields a module version of the bijection betweenthe set of localizing subcategories of the derived categoryof R-modules and the set of subsets of Spec R which was givenby Neeman.     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

Coherent Functors and Covariantly Finite Subcategories

Given a locally presentable additive category A, we study a class of covariantly finite subcategories which we call definable. A definable subcategory arises from a set of coherent functors F _i on A by taking all objects X in A such that F _i X=0 for all i. We give various characterizations of definable subcategories, demonstrating that all covariantly finite subcategories which arise in practice are of this form. This is based on a filtration of the category of all coherent functors on A.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov–Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category ??????(X) of quasi‐coherent sheaves on X is such a category and so has these features. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gorenstein-Projective Modules over T_（m,n）（A）

A new class of Gorenstein algebras T _m,n (A) is introduced, their module categories are described, and all the Gorenstein-projective T _m,n (A)-modules are explicitly determined.     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

Geometry of spaces of compact operators

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if for all reflexive Banach spaces Y. We show that X ^* has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.     

The Jacobian of a nonorientable Klein surface

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surfaceY is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms onY quotiented by the torsion-free part of the first integral homology ofY. Denote byX the double cover ofY given by orientation. The Jacobian ofY is identified with the space of all degree zero holomorphic line bundlesL overX with the property thatL is isomorphic to σ*/-L, where σ is the involution ofX.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

Gorenstein Analogue of Auslander's Theorem on the Global Dimension

In this paper, we prove that the Gorenstein analogue of the well-known Auslander's theorem on the global dimension holds true. Namely, we prove that the Gorenstein global dimension of a commutative ring R is equal to the supremum of the set of Gorenstein projective dimensions of all cyclic R-modules.     

A Hereditary Torsion Theory for Modules Over Integral Domains and Its Applications

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.     

On Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G ₁ → G ₀ → G ⁰ → G ¹ → … such that the complex Hom _R (G, H) is exact for each projective left R-module H, the module Im(G ₀ → G ⁰) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.     

Exceptional bundles on nodal Enriques surfaces

In this paper we prove that an Enriques surfaceX has a smooth rational curve if and only if there exists an exceptional bundleE _t of rank 2 withc ₂ (E _t )=t for any integer t onX. We describe all exceptional bundles of rank 2 on Enriques surfaces and show that they are all stable with respect to any ample divisor.     

An approximation theorem for sections of reflexive sheaves

 We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map has dense image, too. We also characterize the reflexivity of a torsion-free coherent sheaf on complex manifolds in terms of absolute gap sheaves or Kontinuit?tssatz. Received: 14 September 2001 / Revised version: 29 January 2002     

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of Cp. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

Vanishing thetanull and hyperelliptic curves

Let ??_g,2 be the moduli space of curves of genus g with a level‐2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ??_6,2. We prove also that for all g ≥ 3, each component of the hyperelliptic locus in ??_g,2 is a connected component of the intersection of g – 2 thetanull divisors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Feller's One-Dimensional Diffusions as Weak Solutions to Stochastic Differential Equations

A continuous strong Markov process X on the line generated by Feller's generalized second order differential operator D_mD is considered. Supposed that the canonical scale p is locally the difference of two bounded convex functions, that the speed measure m contains a strictly positive absolutely continuous component, and that both boundaries of the state space R are inaccessible. Then the process X is characterized as a weak solution to a stochastic differential equation involving local time.