We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given `geometric' invariant as a property of its sites of de nition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to speci c invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equiva...

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions – Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of co...

It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of...

We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules. We also con...

Given a graph G = ([n], E) and w ∈ R , consider the integer program maxx∈{±1}n ∑ ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijv T i vj , where the maximum is taken over all unit vectors vi ∈ R. The integrality gap of this relaxation is known as the Grothendieck constant κ(G) of G. We present a closed-form formula for the Grothendieck constant of K5-minor free...

Let K be a field of characteristic pi^O, to exclude trivial cases) and let G be a finite goup. A XG-module M is a finite dimensional K-vector space, on which G acts Jf-linearly from the left. The Green ring a(G) of G(w.r.t. K) is the free abelian group, spanned by the isomorphism classes of indecomposable i£G-modules, with the multiplication induced from the tensor product © x of KGmodules (see...

For any separated map f : X → Y of quasi-compact quasiseparated schemes, Rf∗ : D + qc(X) → D (Y ) has a right adjoint f . If f is proper and pseudo-coherent (e.g., finitely-presented and flat) then Duality and tor-independent Base Change hold for f . Preface This is a research summary written early in 1991, concerning results obtained by the author during a stay at MSRI in Berkeley during 1989–...

It is well known that interpolation properties of logics underlying speci.cation formalisms play an important role in the study of structured speci.cations, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multi-logic sp...

By a ‘completion’ on a 2-category K we mean here an idempotent pseudomonad on K . We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of V-Cat for V a symmetric monoidal closed category, with the Grothendieck completion [7], defined in the setting of S-Indexed Cat for S a to...

We relate closure operations for ideals and for submodules to non-flat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and...