Universal deformation rings and cyclic blocks
Let \(\Gamma\) be a profinite group, \(k\) an algebraically closed field, \(V\) a \(k\Gamma\)-module and \(W=W(k)\) the ring of infinite Witt vectors over \(k\). If \(\text{End}_{k\Gamma}(V)=k\), then there is a topological ring \(R(\Gamma,V)\) which is universal for the deformations of \(V\) over local \(W\)-algebras \(A\) which are the projective limit of their discrete Artinian quotients. In this paper the authors consider the case when \(\Gamma=G\) is finite. Let \(V\) be a module belonging to the block \(B\) of \(kG\) having defect group \(D\) of order \(p^d\), and denote \(d^j(V)=\dim_k\widehat H^j(G,\text{Hom}_k(V,V))\), where \(\widehat H^j\) is the \(j\)-th Tate cohomology group. The main result of the paper reads as follows. Assume that \(D\) is cyclic, \(d^0(V)\leq 1\), and let \(e\) be the number of simple \(B\)-modules. Then \(d^j(V)\leq 1\) for all \(j\geq 0\), and one has the following cases: (1) \(d^1(V)=0\) if and only if \(R(G,V)=W/(d^2(V)\cdot p^dW)\). (2) \(d^1(V)=0\) if and only if \(R(G,V)\simeq\text{Inv}_E(WD)/Ws\), where \(W\) is the unique cyclic group of automorphisms of \(D\) of order \(e\), \(s=0\) if \(e=1\), and \(s=\sum_{x\in D}x\) if \(e>1\). Several examples and applications are also given, especially towards a Galois theoretical conjecture of Dèbes and Deschamps.
scientific article; zbMATH DE number 1552934