We apply the methods of Căldăraru to construct a twisted FourierMukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its ‘mirror’ partner. As corollaries, we show that the two spaces are connected by a one-parameter famil...

We clarify the role of Hofmann’s Axiom in the old-style definition of a semiabelian category. By removing this axiom we obtain the categorical counterpart of the notion of an ideal-determined variety of universal algebras – which we therefore call an ideal-determined category. Using known counter-examples from universal algebra we conclude that there are idealdetermined categories which fail to...

LetM be a tensor category with coefficients in a fieldK of characteristic 0, that is, a K-linear pseudo-abelian symmetric monoidal category such that the tensor product ⊗ of M is bilinear. Then symmetric and exterior powers of an object M ∈ M make sense, by using appropriate projectors relative to the action of the symmetric groups on tensor powers of M . One may therefore introduce the zeta fu...

An analogue of the Fourier transform is developed for D-algebras. G. Laumon’s equivalence between the derived category of D-modules on an abelian variety and the derived category of O-modules on the universal extension of the dual variety is seen as a degenerate case of a duality for twisted differential operators (tdo’s) with respect to which the dual of a generic tdo is again a tdo.

There is therefore little which is original contained in these pages. I have given a more or less complete proof of Murre’s result in §4, so as to make the comparison between the different decompositions in 5.3. Also included is a proof (3.5) of the unsurprising fact that the category of motives constructed using rational equivalence of cycles is in general not an abelian category. Otherwise pr...

For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules [11] are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation equivalence between Mod(O) and Qch(O).

1.1 Fix a variety X over C. The Riemann-Hilbert correspondence identifies the category of perverse sheaves on X(C) with the (abelian) category of regular holonomic D-modules on X. This is a remarkable and deep theorem in the theory of linear partial differential equations. In this note we will investigate this correspondence in simple examples, exploring the topological and algebraic sides as w...

Let D be a triangulated category with a cluster tilting subcategory U . The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U , and that, in turn, support tilting subcategories of D/U can be lifted uniquely to weak cluster tilting subcategories of D.

The starting point for the ideas in this paper is a result of Kevin Burke [2, Prop 3.2.5] in the model theory of modules. This result establishes a correspondence between syntactically defined “pp-imaginaries” and certain, coherent, functors. Specifically, fix a ring R and let R-Mod be the category of left R-modules and R-mod be the category of finitely presented left R-modules. The category of...

We construct two abelian extensions of Ω3G, one by the abelian group Map(D4 0G,U(1)) and the other by Map((D )0G,U(1)), where D4 0G is the group of mappings from the unit four-disc D 4 to G = SU(N), N ≥ 3, that are equal to 1 on the boundary S3, and the prime indicates the opposite orientation of D4. They are homotopically equivalent to a U(1) principal bundle over Ω3G and the associated line b...