Cat-valued sheaves

Let \(\mathcal{\widetilde{O}}\)(B) be the category of (open) subcategories of a topological groupoid B: In this paper we study Cat-valued sheaves over category \(\mathcal{\widetilde{O}}\)(B): The paper introduces a notion of categorical union, such that the categorical union of subcategories is a subcategory. We use this definition of categorical unions to define a categorical cover of a topological category. Instead of assuming a Grothendieck topology, we define Cat-valued sheaves in terms of the categorical cover defined in this paper. The main result is the following. For a fixed category C, the categories of local functorial sections from B to C define a Catvalued sheaf on \(\mathcal{\widetilde{O}}\)(B): Replacing C with a categorical group G; we find a CatGrp-valued sheaf on \(\mathcal{\widetilde{O}}\)(B): We also relate and distinguish our construction with the notion of stacks.     

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction. Received: 19 February 1998 / Revised version: 28 December 1998     

Gorenstein injective sheaves

We define Gorenstein injective quasi-coherent sheaves, and prove that the notion is local in case the scheme is Gorenstein. We also give a new characterization of a Gorenstein scheme in terms of the total acyclicity of every acyclic complex of injective quasi-coherent modules.     

Toric co-Higgs sheaves

We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.     

Quantales and their sheaves

A quantale is a complete lattice provided with a binary multiplication which, in particular, distributes over in each variable. First we study the quantales, their morphisms, their points and the relations with the theory of locales: the case where the multiplication is just the meet. We describe the presheaves and sheaves on a quantale and exhibit a link between the corresponding toposes and quantum logic.Research supported by NATO grant 476/84.     

Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.