We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois desce...

In this paper, we introduce the notion of Grothendieck enriched categories for over a sufficiently nice monoidal category $$\mathcal {V}$$ , generalizing classical categories. Then establish Gabriel-Popescu type theorem We also prove that property being is preserved under change base by right adjoint functor. particular, if take as complexes abelian groups, obtain dg As an application main resu...

This problem was solved in the negative by Bridson and Grunewald in [6] who produced many examples of groups G, and proper subgroups H for which û an isomorphism. The method of proof of [6] was a far reaching generalization of an example of Platonov and Tavgen [33] that produced finitely generated examples that answered Grothendieck’s problem in the negative (see also [5]). Henceforth, we will ...

∂if = f− sif xi − xi+1 where si acts on f by transposing xi and xi+1 and let π̃i = ∂i(xi(1− xi+1)f) Then the Grothendieck-Demazure polynomial κα, which is attributed to A. Lascoux and M. P. Schützenberger, is defined as κα = x α1 1 x α2 2 x α3 3 ... if α1 ≥ α2 ≥ α3 ≥ ..., i.e. α is non-increasing, and κα = π̃iκαsi if αi < αi+1, where si acts on α by transposing the indices. Example 2.1. Let α = (...

Fulton and MacPherson introduced the notion of bivariant theories related to Riemann-Roch-theorems, especially in the context of singular spaces. This is powerful formalism, which is a simultaneous generalization of a pair of contravariant and covariant theories. Natural transformations of bivariant theories are called Grothendieck transformations, and these generalize a pair of ordinary natura...

How can one prove a sharp inequality? Symmetrization, Fourier analysis, and probability are often used, and we will survey some of these methods through examples. We then survey sharp constants in Grothendieck inequalities, leading to some recent work on computing the best constant for a Grothendieck-type inequality of Khot and Naor. (Joint work with Aukosh Jagannath and Assaf Naor)

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic DonaldsonThomas theory.

In the Grothendieck cohomology of a flag variety G/H there are two canonical additive bases, namely, the Demazure basis [D] and the Grothendieck basis [KK]. We present explicit formulae that reduce the multiplication of these basis elements to the Cartan numbers of G.

In each dimension n ≥ 3 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.