In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results...

We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Our formula is stated in terms of coefficients that are uniquely determined by the geometry and can be computed by an explicit combinatorial algorithm. We conjecture that these coefficients have signs t...

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves wi...

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich’s motivic integration via arc schemes. In view of i...

Grothendieck fibrations have played an important role in homotopy theory. Among others, theywereused byThomason to describehomotopy colimits of small categories and byQuillen to derive long exact sequences of higher K-theory groups. We construct simplicial objects, namely the fibred and the cleaved nerve, to characterize the homotopy type of a Grothendieck fibration by using the additional stru...

This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson’s theory-space where “branes” are still branes. In this work, we focus on D-strings and construct the moduli stack of morphisms from Azumaya prestable curves C with a fundamental module E to a fixed target Y of a given combinatorial type. Such a morphism give...

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family of model structures on the fibers $\mathcal E_A$ and on the basis category $\mathcal B$ combines into a model structure on the total category $\mathcal E$, ...

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux’s interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this ...

We generalize the notion of a jet scheme (truncated arc space) to arbitrary fat points via adjunction, and show that this yields for each fat point, an endomorphism on each schemic Grothendieck ring as defined in [17]. We prove that some of the analogues for linear jets still hold true, like locally trivial fibration over the smooth locus. In this formalism, we can define several generating zet...

Define a truncation rt(p) of a polynomial p in {x1, x2, x3, . . .} as the polynomial with all but the first t variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in K(Flags(C)), in particular gener...