6,790 research outputs found
Representations and Cohomology of finite group schemes
The article covers developments in the representation theory of finite group
schemes over the last fifteen years. We start with the finite generation of
cohomology of a finite group scheme and proceed to discuss various consequences
and theories that ultimately grew out of that result. This includes the theory
of one-parameter subgroups and rank varieties for infinitesimal group schemes;
the -points and -support spaces for finite group schemes, modules of
constant rank and constant Jordan type, and construction of bundles on
projective varieties associated with cohomology ring of an infinitesimal group
scheme . In the last section we discuss varieties of elementary subalgebras
of modular Lie algebras, generalizations of modules of constant Jordan type,
and new constructions of bundles on projective varieties associated to a
modular Lie algebra.Comment: 31 pag
Towards Donovan's conjecture for abelian defect groups
We define a new invariant for a -block, the strong Frobenius number, which
we use to address the problem of reducing Donovan's conjecture to normal
subgroups of index p. As an application we use the strong Frobenius number to
complete the proof of Donovan's conjecture for 2-blocks with abelian defect
groups of rank at most 4 and for 2-blocks with abelian defect groups of order
at most 64
Morita equivalence classes of 2-blocks of defect three
We give a complete description of the Morita equivalence classes of blocks
with elementary abelian defect groups of order 8 and of the derived
equivalences between them. A consequence is the verification of Brou\'e's
abelian defect group conjecture for these blocks. It also completes the
classification of Morita and derived equivalence classes of 2-blocks of defect
at most three defined over a suitable field
Derived induction and restriction theory
Let be a finite group. To any family of subgroups of ,
we associate a thick -ideal of the
category of -spectra with the property that every -spectrum in
(which we call -nilpotent) can be
reconstructed from its underlying -spectra as varies over .
A similar result holds for calculating -equivariant homotopy classes of maps
into such spectra via an appropriate homotopy limit spectral sequence. In
general, the condition implies strong
collapse results for this spectral sequence as well as its dual homotopy
colimit spectral sequence. As applications, we obtain Artin and Brauer type
induction theorems for -equivariant -homology and cohomology, and
generalizations of Quillen's -isomorphism theorem when is a
homotopy commutative -ring spectrum.
We show that the subcategory contains many
-spectra of interest for relatively small families . These
include -equivariant real and complex -theory as well as the
Borel-equivariant cohomology theories associated to complex oriented ring
spectra, any -local spectrum, the classical bordism theories, connective
real -theory, and any of the standard variants of topological modular forms.
In each of these cases we identify the minimal family such that these results
hold.Comment: 63 pages. Many edits and some simplifications. Final version, to
appear in Geometry and Topolog
Classifying blocks with abelian defect groups of rank for the prime
In this paper we classify all blocks with defect group up to Morita equivalence. Together with a recent paper of Wu,
Zhang and Zhou, this completes the classification of Morita equivalence classes
of -blocks with abelian defect groups of rank at most . The
classification holds for blocks over a suitable discrete valuation ring as well
as for those over an algebraically closed field. The case considered in this
paper is significant because it involves comparison of Morita equivalence
classes between a group and a normal subgroup of index , so requires novel
reduction techniques which we hope will be of wider interest. We note that this
also completes the classification of blocks with abelian defect groups of order
dividing up to Morita equivalence. A consequence is that Broue's abelian
defect group conjecture holds for all blocks mentioned above
Groups elementarily equivalent to a free nilpotent group of finite rank
In this paper we give a complete algebraic description of groups elementarily
equivalent to a given free nilpotent group of finite rank
- …