Stable equivalences of adjoint type
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Changchang Xi
Abstract
In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Broué's conjecture was made (see for instance [Broué M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 1–26]). The main results in this paper are: Let A and B be two artin k-algebras such that A and B are projective over k, and let Hn(A) and Hn(B) be the n-th Hochschild cohomology groups of A and B, respectively. (1) If A and B are stably equivalent of adjoint type, then Hn(A) ≃ Hn(B) for all n ≥ 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type.
2000 Mathematics Subject Classification: 16G10, 16E30; 16G70, 18G05, 20J05.
© Walter de Gruyter
Articles in the same Issue
- q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
- Antifinitary linear groups
- Comparing GLn-representations by characteristic-free isomorphisms between generalized Schur algebras
- Stable equivalences of adjoint type
- A partial solution of the isoperimetric problem for the Heisenberg group
- Brill-Noether problems in higher dimensions
- Second main theorem with truncated counting function in several complex variables for moving targets
- Semifields with free automorphism groups
Articles in the same Issue
- q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
- Antifinitary linear groups
- Comparing GLn-representations by characteristic-free isomorphisms between generalized Schur algebras
- Stable equivalences of adjoint type
- A partial solution of the isoperimetric problem for the Heisenberg group
- Brill-Noether problems in higher dimensions
- Second main theorem with truncated counting function in several complex variables for moving targets
- Semifields with free automorphism groups
