On a Question of Rickard on Tensor Products of Stably Equivalent Algebras

Abstract

Let r be a positive integer, let p be a prime, and denote an algebraic closure of the prime field . After observing that the principal block B of is stably equivalent of Morita type to its Brauer correspondent b, we compute the radical series of the center Z(b), and, using GAP, the radical series of Z(B) in the cases p^r ∈ {3, 4, 5, 7, 8}. In these cases, the dimensions of the last nonzero power of the radical of Z(b) and Z(B) are different, and it follows that the algebra is not stably equivalent of Morita type to . This yields a negative answer to a question of Rickard.

KEYWORDS:

• stable equivalence
• tensor product
• self-injective algebra

2000 AMS Subject Classification:

• Primary 16E35
• 18E30
• 20C05

Acknowledgment

The idea of this article was born during a visit of both authors in Beijing Normal University. We are very grateful to Yuming Liu for his great hospitality. Moreover, we thank Yuming Liu for suggesting a question to us leading to the present article, and also for pointing out that it is not sufficient to show an abstract nonisomorphism of the centers of the blocks. This led us to complete our proof by adding Lemma 9.

Funding

Both authors were supported by a grant STIC Asie ’Escap’ from the Ministère des Affaires Étrangères de la France.