Hilbert-Samuel function and Grothendieck group

An algorithm for the Hilbert - Samuel function of a primary

The Grothendieck Group

In the red corner, topological K-theory! The study of stable equivalences of vector bundles over a topological space, and the engine behind Bott periodicity, a result whose reverberations are felt throughout algebraic topology. In the blue corner, modular representation theory! The story of representations in positive characteristic, where the CDE triangle powers applications in group theory an...

Hilbert-samuel Polynomials for the Contravariant Extension Functor

Let (R, m) be a local ring and M and N finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ExtR(M, N/m nN). A number of corollaries are given and more general filtrations are also considered.

Grothendieck Groups of Abelian Group Rings

Let R be a noetherian ring, and G(R) the Grothendieck group of finitely generated modules over R. For a finite abelian group n, we describe G(Rn) as the direct sum of groups G(R’). Each R’ is the form RI<,,, I/n], where n is a positive integer and Cn a primitive nth root of unity. As an application, we describe the structure of the Grothendieck group of pairs (H. u), where His an abelian group ...

The Grothendieck Group of Hopf Algebras

Let H be a cosemisimple Hopf algebra over an algebraically closed field k. Assume that H contains a simple subcoalgebra of dimension 9. We show that if H has no simple subcoalgebras of even dimension then H contains either a grouplike element with order 2 or 3, a Hopf subalgebra of dimension 75, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. I...