We give a nonrecursive combinatorial formula for the expansion of a stable Grothendieck polynomial in the basis of stable Grothendieck polynomials for partitions. The proof is based on a generalization of the EdelmanGreene insertion algorithm. This result is applied to prove a number of formulas and properties for K-theoretic quiver polynomials and Grothendieck polynomials. In particular we for...

The little Grothendieck problem consists of maximizing Σ ij Cijxixj for a positive semidef-inite matrix C, over binary variables xi ∈ {±1}. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ ℝ dn × dn a positive semidefinite matrix, the objective is to maximize [Formula: see text] restricting Oi to take values...

HomX denotes the sheaf-Hom functor of OX -complexes: HomX(E,F )(U) := Hom • U (E|U , F |U ) (U ⊂ X open), the restriction map for U ′ ⊂ U being the obvious one. This “dynamic” sheafified version of Hom• has a derived functor RHom•, defined as usual via q-injective resolutions (which always exist!). Similarly, we have a sheaf-theoretic version of ⊗, and its left-derived functor ⊗ = , defined via...

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point. At the Edinburgh meeting on the model theory ...

Abstract In 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams schemes, Lecture Notes Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the product a decade’s work, giving definitive, state-of-the-art treatment duality. this article, we achieve sharp improvement: begin by new proof base-change theorem, which can handle unbounded complex...

Themain purpose of this note is to present a characterization of rational singularities in characteristic 0. The essence of the characterization is that it is enough to require less than the usual definition. Theorem 1. Let φ : Y → X be a morphism of varieties over C, and let ρ : X → Rφ∗ Y be the associated natural morphism. Assume that Y has rational singularities and there exists a morphism (...

A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was gi...

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories. 1. Introduction. Given a Grothendieck category Ꮽ, we can associate with it the lattice of all localizing categories of Ꮽ and denote it by Tors(Ꮽ). We ...

Quantum K-theory is a K-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory of the classical flag variety Fln, we define and study quantum Grothendieck polynomials. We conjecture that they represent Schubert classes (i.e., the natural basis elements) in the quantum K-theory of Fln, and present strong evidence for thi...

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.