The Grothendieck Group of G-Equivariant Modules Over Coordinate Rings of G-Orbits (original) (raw)

Desingularizations of varieties of nullforms

Inventiones Mathematicae, 1979

1.1 Let G be a connected reductive group over an algebraically closed field k. Let V be a pointed affine G-variety. So G acts on V and the base point * of V is

Grothendieck Groups of Invariant Rings: Filtrations

Proceedings of the London Mathematical Society, 1993

We investigate the Grothendieck group G a (R) of finitely generated modules over the ring of invariants R = S G of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G 0 (R) that is defined in terms of (Gabriel-Rentschler) Krull dimension, properties of G 0 (R) are derived to a large extent from the connections between the sets of prime ideals of 5 and R. A crucial ingredient is an equivalence relation ~ on Spec/? that was introduced by Montgomery [25]. For example, we show that

Grothendieck Groups of Quotient Singularities

Transactions of the American Mathematical Society, 1992

The Grothendieck group of S[G] carries a natural structure of a ring, isomorphic to G0(C[<7]). We show how the structure of Go(R) is related to the structure of the ramification locus of V over V¡G , and the action of G on it. The first connection is given by showing that Ker y/ is the ideal generated by [C] if and only if G acts freely on V. That this is sufficient has been proved by Auslander and Reiten in [4]. To prove the necessity we show the following: Let U be an integrally closed domain and T the integral closure of U in a finite Galois extension of the field of quotients of U with Galois group G .

Noncommutative desingularization of orbit closures for some representations of GLnGL_nGLn

arXiv: Algebraic Geometry, 2012

We describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.

On the geometry of extensions of irreducible modules for simple algebraic groups

1987

Let G be a simple, simply connected affine algebraic group over an algebraically closed field k of non-zero characteristic p. We consider the problem of determining the extensions of irreducible modules by irreducible modules. The extensions may be realized as submodules of modules induced from characters on a Borel subgroup of G. The geometry of the distribution of composition factors of those induced modules is determined by an operation (namely, alcove transition) of the Weyl group on the space of weights. Genetically in the lowest /? 2-alcove, that operation stabilizes a canonical subset of the set of highest weights of those irreducible modules which extend the irreducible module of some fixed highest weight. The stability leads to an upper bound on that subset, which can be refined using the translation principle. We give a conjecture for the generic distribution of extensions of irreducible modules by a fixed irreducible module. Introduction. Let G be a simple, simply connected affine algebraic group over an algebraically closed field k of non-zero characteristic p. (B,T) is a fixed Borel subgroup and maximal torus pair, X(B) is the character group of B, and 2? opp is the opposite Borel subgroup. Take the positive roots of (G, T) to be the roots of (£ opp , T). Let G x be the kernel of the Frobenius morphism of G. Let {^(χ)}?!™ 07 * = {H^G/B, L(χ))}f% G/B be the sheaf cohomology modules of the homogeneous space G/B at the line bundle L(χ) induced from a character χ on B. For each dominant character λ, H°(λ) has as its socle the irreducible module M λ of highest weight λ. The formal character of M λ can be computed in terms of the formal characters of the modules {M μ \ μ Φ λ, μ strongly linked to λ}, once the multiplicities [H°(λ): M μ ] of the M μ as composition factors of H°(\) are known. Let X[H°(λ)]= {(μ,«) e X(B)XZ >Q \[H°(λ): M μ ] = n}, let λ\= {μ ^ X(B)\[H°(λ): M μ ]Φ