The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault turns out to be canonically isomorphic with local (relative) Grothendieck and Sato so that it provides a handy way representing latter. In this paper we use give simple explicit expressions hyperfunctions, some fundamental operations on them related duality theorems. This approach also yields new insight ...

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...

We give an exposition and generalization of Orlov’s theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this ...

In this article, we study how the Grothendieck group of coherent sheaves can be used to describe D-branes. We show how global bound state construction in topological K-theory can be adapted to our context, showing that D-branes wrapping a subvariety are holomorphically classified by a relative K-group. By taking the duality between the relative K-groups and the K-homologies, we show that D-bran...

We study monoidal categorifications of certain subcategories C J $\mathcal {C}_J$ finite-dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings K ( ) $K(\mathcal {C}_J)$ are closely related to the category quiver Hecke type A ∞ $A_\infty$ via generalized Schur–Weyl duality functors. In particular, when is $A$ or B $B$ , subcategory coincid...

We pursue the principles of duality and symmetry building upon Pratt’s idea of the Stone Gamut and Abramsky’s representations of quantum systems. In the first part of the paper, we first observe that the Chu space representation of quantum systems leads us to an operational form of state-observable duality, and then show via the Chu space formalism enriched with a generic concept of closure con...

The goal of this paper is to give an account of classical Tannaka duality [C⁄] in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments [⁄SR, DM⁄] and quantum groups [⁄D1⁄]. Expertise in neither representation theory nor category theory is assumed. Naively speaking, Tannaka duality theory is the study of the interplay which...

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One main varieties satisfy rational Poincare ...

In this paper, we establish that Gelfand duality holds between the category of commutative C*-algebras and the category of compact, completely regular locales in any Grothendieck topos. It should be remarked immediately that this result represents the final step in a chain of preliminary papers that have appeared over a period of time. Indeed, the work contained in these papers was originally p...

Some important examples and explications of Grothendieck duality which should have been included in Chapter 5 of [C1] were overlooked. Several such topics are treated here. One question which can be asked about the trace map H(X,ΩX/k) → k for a proper smooth scheme X of pure dimension n over a field k is how it relates to the trace map for a smooth divisor. Generalizing a little, one can ask wh...