On the representation dimension of rank 2 group algebras and related algebras
نویسندگان
چکیده
The representation dimension was defined by M.Auslander in 1970 and is, due to spectacular recent progress, one of the most interesting homological invariants in representation theory. The precise value is not known in general, and is very hard to compute even for small examples. For group algebras, it is known in the case of cyclic Sylow subgroups, due to Auslander’s fundamental work. For some group algebras (in characteristic 2) of rank at least 3 the precise value of the representation dimension follows from recent work of R.Rouquier. There is a gap for group algebras of rank 2; here the deep geometric methods do not work. In this paper we show that for all n ≥ 0 and any field k the commutative algebras k[x, y]/(x, y) have representation dimension 3. For the proof, we give an explicit inductive construction of a suitable generator-cogenerator. As a consequence, we obtain that the group algebras in characteristic 2 of the groups C2 × C2m have representation dimension 3. Note that for m ≥ 3 these group algebras have wild representation type. MSC Classification: 16G10 (primary); 13D05, 16S50, 20C05 (secondary).
منابع مشابه
A note on essentially left $phi$-contractible Banach algebras
In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fou...
متن کاملCrossed Products by Finite Group Actions with the Rokhlin Property
We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: • AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. • Simple unital AH algebras with slow dimension growth and real rank zero. • C*-algebras with real rank ...
متن کاملOn nuclei of sup-$Sigma$-algebras
In this paper, algebraic investigations on sup-$Sigma$-algebras are presented. A representation theorem for sup-$Sigma$-algebras in terms of nuclei and quotients is obtained. Consequently, the relationship between the congruence lattice of a sup-$Sigma$-algebra and the lattice of its nuclei is fully developed.
متن کاملOn p-semilinear transformations
In this paper, we introduce $p$-semilinear transformations for linear algebras over a field ${bf F}$ of positive characteristic $p$, discuss initially the elementary properties of $p$-semilinear transformations, make use of it to give some characterizations of linear algebras over a field ${bf F}$ of positive characteristic $p$. Moreover, we find a one-to-one correspondence between $p$-semiline...
متن کامل