We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite st...

We prove a sheaf-theoretic derived-category generalization of GreenleesMay duality (a far-reaching generalization of Grothendieck’s local duality theorem): for a quasi-compact separated scheme X and a “proregular” subscheme Z—for example, any separated noetherian scheme and any closed subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived co...

The \generalized universe of sets" aspect of toposes is relatively easy to understand and is well documented in the literature: start with Goldblatt [1] and proceed via Mac Lane and Moerdijk [5], or MacLarty [4], to Johnstone [2]. The basic trick is to use categorical properties to characterize set-theoretic constructions in the category of sets, and thence to transfer them to other categories ...

In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a generalisation of a Littlewood-Richardson rule for Grothendieck polynomials.

— We show how the ideas of Leray (sheaf theory), Grothendieck (derived categories) and Sato (microlocal analysis) lead to the microlocal theory of sheaves which allows one to reduce many problems of linear partial differential equations to problems of microlocal geometry. Moreover, sheaves on Grothendieck topologies are a natural tool to treat growth conditions which appear in Analysis. Résumé ...

A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological...

In 1949, Andre Weil published striking conjectures linking number theory to topology and a striking strategy for a proof [Weil, 1949]. Around 1953, Jean-Pierre Serre took on the project and soon recruited Alexander Grothendieck. Serre created aseries of concise elegant tools which Grothendieck and coworkers simplified into thousands of pages of category theory. Some have complained of this styl...