We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. also transformations and formulas generalize the Adams-Riemann-Roch equivariant localization theorems. As applications, we exhibit projective toric variety $X$ whose of vector bundles does not surject onto its ordinary $K...

In the setting of Banach lattices weak (resp. positive) Grothendieck spaces have been defined. We localize such notions by defining new classes sets that we study and compare with some quite related different classes. This allows us to introduce corresponding linear operators.

In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between (Banach for which weak- weak*-sequential convergences in the dual space coincide) measures. This was developed further book with J. Uhl Jr. Vector Additionally, Diestel’s included section several open problems about structural properties of spaces, only half them have been solved ...

Let I be a small indexing category, G : I → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (re...

Alexander Grothendieck was born in Berlin on 28 March 1928. His father, Sascha Shapiro, an anarchist originally from Russia, took an active part in the revolutionary movements first in Russia, and then in Germany, during the 1920s, where he met Hanka Grothendieck, Alexander’s mother. After the Nazis came to power in Germany it was too dangerous for a Jewish revolutionary to stay there, and the ...