On topological invariants of stratified maps with non-Witt target

The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.

     

Exterior Tensor Product of Perverse Sheaves

Under certain assumptions, we prove that the Deligne tensor product of the categories of constructible perverse sheaves on pseudomanifolds X and Y is the category of constructible perverse sheaves on X×Y. The functor of the exterior Deligne tensor product is identified with the exterior geometric tensor product.     

Resolution of unbounded complexes in Grothendieck categories

As Spaltenstein showed, the category of unbounded complexes of sheaves on a topological space has enough K-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives.     

The differential Galois theory of strongly normal extensions

Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product.

We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms.

This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields.

On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.

     

Locally finitely presented categories of sheaves

Let A be a locally finitely presented Grothendieck category. It is shown that a class of localizations of A in the sense of Bousfield is again locally finitely presented. The criterion is applied to torsion-free classes in A, sheaves and separated presheaves on a generalized ringed space, and representations of partially ordered sets.     

Tensor product varieties,perverse sheaves,and stability conditions

The space spanned by the class of simple perverse sheaves in Zheng (2008) without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable highest weight modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. The two stability conditions coincide with the localization process in Zheng (2008), by using supports and singular supports of complexes of sheaves, respectively.     

Invariants de Von Neumann des faisceaux analytiques cohérents

In order to study the group of holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf on a n-dimensionnal smooth projective variety and each Galois infinite unramified covering , whose Galois group is denoted by , cohomology groups denoted by are attached, such that: 1. The underly a cohomological functor on the abelian category of coherent analytic sheaves on X. 2. If is locally free, is the group of holomorphic sections of the pull-back to of the holomorphic vector bundle underlying . 3. belongs to a category of -modules on which a dimension function with real values is defined. 4. Atiyah's index theorem holds [Ati76]: The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96] of an abelian category containing every closed -submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods to study Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar r?le here. We first construct a -cohomology theory () for coherent analytic sheaves on a complex space endowed with a proper action of a group such that conditions 1-2 are fulfilled. The -cohomology on the Galois covering of a coherent analytic sheaf onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of by the locally constant sheaf on X associated to the left regular representation of the discrete group in the space of functions on . Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category of and are a good model of the Čech complex calculating -cohomology. Using this we prove that , if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title of the article, Betti numbers and Novikov-Shubin invariants of are the Von Neumann invariants of the coherent analytic sheaf . We also make the connection with Atiyah's -index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.

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Received: 30 October 1998 / Published online: 8 May 2000     

Homological Algebra of Twisted Quiver Bundles

Several important cases of vector bundles with extra structure(such as Higgs bundles and triples) may be regarded as examplesof twisted representations of a finite quiver in the categoryof sheaves of modules on a variety/manifold/ringed space. Itis shown that the category of such representations is an abeliancategory with enough injectives by the construction of an explicitinjective resolution. Using this explicit resolution, a longexact sequence is found that computes the Ext groups in thisnew category in terms of the Ext groups in the old category.The quiver formulation is directly reflected in the form ofthe long exact sequence. It is also shown that under suitablecircumstances, the Ext groups are isomorphic to certain hypercohomologygroups.     

某些半群子范畴中的张量积

半群范畴S中张量积首先在中引入。T∈ob S称为A,B∈ob S的张量积(记为AB),如果存在双同态t:A×B→T(相当于中线性平衡映射),且对于任意双同态s:A×B→C∈ob S总存在唯一的同态μ:T→C,使s-ut。确认了张量积的存在唯一。等引入交换半群、半格等子范畴中的张量积,其定义与上述基本相同,仅将S改为该子范畴,此外该划了一些半群类的张量积。本文在§1从任意半群簇V中张量积与其在S中     

Rosenberg’s reconstruction theorem

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.     

Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds

We apply the methods of C a?ld?raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds which are fibred by Lagrangian abelian surfaces. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain Lagrangian fibration and the derived category of twisted sheaves on its ‘mirror’ partner. As a corollary, we extend the original Fourier-Mukai transform to degenerations of abelian surfaces. Another consequence of the general theory is that the holomorphic symplectic four-fold and its mirror are connected by a one-parameter family of deformations through Lagrangian fibrations.     

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P(z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P(z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P(z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.     

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of $\mathbb{Q}$-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.     

Cohomological property of vector bundles on biprojective spaces

This paper investigates the cohomological property of vector bundles on biprojective space. We will give a criterion for a vector bundle to be isomorphic to the tensor product of pullbacks of exterior products of differential sheaves.

     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.     

Functors of the category of abelian sheaves on regular affine scheme

In this short paper, we prove that ifR is a regular local ring of unequal characteristic, then there exists an additive covariant functorG from the category of abelian sheaves on SpecR to the category of abelian groups such that id _R (G(R))>dimG(R). This result shows that the answer to the question 3.8 (ii) in [3] may be negative.     

A Quillen Model Structure for 2-Categories

We describe a cofibrantly generated Quillen model structure on the locally finitely presentable category 2-Cat of (small) 2-categories and 2-functors; the weak equivalences are the biequivalences, and the homotopy relation on 2-functors is just pseudonatural equivalence. The model structure is proper, and is compatible with the monoidal structure given by the Gray tensor product. It is not compatible with the Cartesian closed structure, in which the tensor product is the product.The model structure restricts to a model structure on the full subcategory PsGpd of 2-Cat, consisting of those 2-categories in which every arrow is an equivalence and every 2-cell is invertible. The model structure on PsGpd is once again proper, and compatible with the monoidal structure given by the Gray tensor product.     

Abelian categories, almost split sequences, and comodules

The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length.

(a) Each indecomposable injective has a simple subobject.

(b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented.

(c) The category has left almost split sequences.

     

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.     

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.