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Wilberd van der Kallen

Publications and source records attributed to Wilberd van der Kallen.

28 records · Page 2Linked to original sources
arXiv2007

A reductive group with finitely generated cohomology algebras

Let $G$ be the linear algebraic group $SL_3$ over a field $k$ of characteristic two. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. We show that the full cohomology ring $H^*(G,A)$ is finitely generated. This extends the finite generation property of the ring of invariants $A^G$. We discuss where the problem stands for other geometrically reductive group schemes.

math.RT↗
arXiv2004

Cohomology with Grosshans graded coefficients

Let the reductive group G act on the finitely generated commutative k-algebra A. We ask if the finite generation property of the ring of invariants A^G extends to the full rational cohomology ring H^*(G,A). We confirm this when G=SL_2 and also when the characteristic of k is large and the action on A is replaced by the `contracted' action on the Grosshans graded ring gr(A). In the printed version of this paper the first level of some spectral sequences was misidentified as the second. This affects twelve subscripts. We have corrected them (in 3.10, 4.11, 4.12).

math.RT↗
arXiv2004

Finite good filtration dimension for modules over an algebra with good filtration

Let G be a connected reductive linear algebraic group over a field k of characteristic p>0. Let p be large enough with respect to the root system. We show that if a finitely generated commutative k-algebra A with G-action has good filtration, then any noetherian A-module with compatible G-action has finite good filtration dimension.

math.RT↗
arXiv2001

Homology stability for unitary groups

In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results.

math.KT↗
arXiv2000

From Mennicke symbols to Euler class groups

Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class group. We suggest that this is the last map in a longer exact sequence of abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the sequence is an orbit set of unimodular two by n matrices over the ring A. If n is at least four we describe a partially defined operation on two by n matrices. We conjecture that this operation describes a group structure on G if A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.

math.KT↗
arXiv1999

Steinberg modules and Donkin pairs

We prove that in positive characteristic a module with good filtration for a group of type E6 restricts to a module with good filtration for a subgroup of type F4. (Recall that a filtration of a module for a semisimple algebraic group is called good if its layers are dual Weyl modules.) Our result confirms a conjecture of Brundan for one more case. The method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, in characteristic not 2, with a computer-aided variation on the old method of Donkin.

math.RT↗
arXiv1996

The Space of Triangles, Vanishing Theorems, and Combinatorics

We consider compactifications of the space of triples of distinct points in projective $n$-space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple. We compute the sections of line bundles on these spaces, and show that they are equal as GL(n) representations to the generalized Schur modules associated to ``bad'' generalized Young diagrams with three rows (Borel-Weil theorem). On the one hand, this yields Weyl-type character and dimension formulas for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis.

alg-geom↗
arXiv1992

On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p

This paper is about sheaf cohomology for varieties (schemes) in characteristic $p>0$. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre's vanishing theorem, but our theorem is based on inverse limits and Grothendieck's theorem on formal functions. The result implies a Grauert--Riemenschneider type theorem.

alg-geom↗