We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare...

let $x$ be a sufficiently nice scheme. we survey some recent progress on dualizing complexes. it turns out that a complex in $kinj x$ is dualizing if and only if tensor product with it induces an equivalence of categories from murfet's new category $kmpr x$ to the category $kinj x$. in these terms, it becomes interesting to wonder how to glue such equivalences.

Grothendieck-Witt spectra represent higher groups and Hermitian K-theory in particular. A description of the spectrum a finite dimensional projective bundle $\mathbb{P}(\mathcal{E})$ over base scheme $X$ is given terms base, using dg category strictly perfect complexes, provided that $\text{Spec }\mathbb{Z}[1/2]$ satisfies resolution property, e.g. if has an ample family line bundles.

This expository article presents a unified ring theoretic approach, based on the theory of Frobenius algebras, to a variety of results on Hopf algebras. These include a theorem of S. Zhu on the degrees of irreducible representations, the so-called class equation, the determination of the semisimplicity locus of the Grothendieck ring, the spectrum of the adjoint class and a non-vanishing result ...

In each characteristic, there is a canonical homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of sets definable in the theory of algebraically closed fields. We prove that this homomorphism is an isomorphism in characteristic zero. In positive characteristics, we exhibit specific elements in the kernel of the corresponding homomorphism of Grothendieck semi rings. Th...

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...

Over a commutative noetherian ring $R$, the prime spectrum controls, via assignment of support, structure both $\mathsf{Mod}(R)$ and $\mathsf{D}(R)$. We show that, just like in $\mathsf{Mod}(R)$, support classifies hereditary torsion pairs heart any nondegenerate compactly generated $t$-structure Moreover, we investigate whether these $t$-structures induce derived equivalences, obtaining new so...

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 ...