46 research outputs found
Large universal deformation rings
We provide a series of examples of finite groups G and mod p representations
V of G whose stable endomorphisms are all given by scalars such that V has a
universal deformation ring R(G,V) which is large in the sense that
R(G,V)/pR(G,V) is isomorphic to a power series algebra in one variable.Comment: 9 pages, 5 figures; for Prop. 2, the description of the groups and
the proof have slightly change
Dihedral blocks with two simple modules
Let be an algebraically closed field of characteristic 2, and let be
a finite group. Suppose is a block of with dihedral defect groups such
that there are precisely two isomorphism classes of simple -modules. The
description by Erdmann of the quiver and relations of the basic algebra of
is usually only given up to a certain parameter which is either 0 or 1. In
this article, we show that if there exists a central extension
of by a group of order 2 together with a block of with
generalized quaternion defect groups such that is contained in the image of
under the natural surjection from onto . As a special
case, we obtain that if for some odd
prime power and is the principal block of .Comment: 11 pages, 5 figures. The arguments work also for non-principal
blocks. The paper has been changed accordingly; in particular, the word
"principal" was removed from the titl
Universal deformation rings and dihedral defect groups
Let k be an algebraically closed field of characteristic 2, and let W be the
ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a
block of kG with dihedral defect group D which is Morita equivalent to the
principal 2-modular block of a finite simple group. We determine the universal
deformation ring R(G,V) for every kG-module V which belongs to B and has stable
endomorphism ring k. It follows that R(G,V) is always isomorphic to a
subquotient ring of WD. Moreover, we obtain an infinite series of examples of
universal deformation rings which are not complete intersections.Comment: 37 pages, 13 figures. Changed introduction, updated reference
Finiteness Theorems for Deformations of Complexes
We consider deformations of bounded complexes of modules for a profinite
group G over a field of positive characteristic. We prove a finiteness theorem
which provides some sufficient conditions for the versal deformation of such a
complex to be represented by a complex of G-modules that is strictly perfect
over the associated versal deformation ring.Comment: 25 pages. This paper is connected to the paper arXiv:0901.010
Universal deformation rings and tame blocks
Let k be an algebraically closed field of positive characteristic, and let W
be the ring of infinite Witt vectors over k. Suppose G is a finite group and B
is a block of kG of infinite tame representation type. We find all finitely
generated kG-modules V that belong to B and whose endomorphism ring is
isomorphic to k and determine the universal deformation ring R(G,V) for each of
these modules.Comment: 14 page
Deformation rings which are not local complete intersections
We study the inverse problem for the versal deformation rings
of finite dimensional representations of a finite group over a
field of positive characteristic . This problem is to determine which
complete local commutative Noetherian rings with residue field can arise up
to isomorphism as such . We show that for all integers
and all complete local commutative Noetherian rings with residue
field , the ring arises in this way. This
ring is not a local complete intersection if , so we
obtain an answer to a question of M. Flach in all characteristics.Comment: 16 page