In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory.
Abstract. 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 W D. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.
Let $k$ be a field, and let $\Lambda$ be a finite dimensional $k$-algebra. We
prove that if $\Lambda$ is a self-injective algebra, then every finitely
generated $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to
$k$ has a universal deformation ring $R(\Lambda,V)$ which is a complete local
commutative Noetherian $k$-algebra with residue field $k$. If $\Lambda$ is also
a Frobenius algebra, we show that $R(\Lambda,V)$ is stable under taking
syzygies. We investigate a particular Frobenius algebra $\Lambda_0$ of dihedral
type, as introduced by Erdmann, and we determine $R(\Lambda_0,V)$ for every
finitely generated $\Lambda_0$-module $V$ whose stable endomorphism ring is
isomorphic to $k$.Comment: 25 pages, 2 figures. Some typos have been fixed, the outline of the
paper has been changed to improve readabilit
Abstract. We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups.
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