Tensor products of higher almost split sequences

Abstract

Abstract. We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama give in [HI11] a criterion for when the tensor product of an n-representation finite algebra and an m-representation finite algebra is (n + m)-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suitable chain map and using a natural notion of tensor product for chain maps.

Tensor products of n-complete algebras

“…The total tensor product of complexes is a functor in a natural way, so we can speak of tensor products of maps of complexes (for a very general treatment of how this is done, see [CE56, IV.4 and IV.5]). An important result which is proved in [Pas17] for d-representation finite algebras is also true for d-complete algebras, namely: Proof. This is proved in the same way as in [Pas17, Section 3.3].…”

“…The following is a result which appeared in [Pas17] in the setting of d-representation finite algebras, and which can be reformulated in the setting of d-complete algebras.…”

“…Notice that the above theorem does not require the algebra A ⊗ B to be (n + m)representation finite (in which case we know a priori that (n + m)-almost split sequences must exist). In the setting of [Pas17], this result is about describing the structure of such sequences. In the setting of d-complete algebras, this result is used to prove that (n + m)-almost split sequences exist, whereas it is a priori not clear that they should.…”

d-Representation-finite self-injective algebras

“…The question thus occurs if there are generalizations of the fundamental results about representation-finite algebras to algebras possessing a d-cluster-tilting module, that is, to d-representation-finite algebras. Significant progess in this and related directions has been made in recent years; see, for example, [5,25,28,29,31,32,39,40,41,43,44,45,46,50,52,53].…”

“…The question thus occurs if there are generalizations of the fundamental results about representation-finite algebras to algebras possessing a d-cluster-tilting module, that is, to d-representation-finite algebras. Significant progess in this and related directions has been made in recent years; see, for example, [5,25,28,29,31,32,39,40,41,43,44,45,46,50,52,53].The fundamental idea of this paper is to construct d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of certain algebras Λ of finite global dimension, called ν d -finite algebras. This class of algebras includes d-representation-finite algebras of global dimension d (which are a higher-dimensional analogue of representation-finite hereditary algebras) and, more generally, twisted fractionally Calabi-Yau algebras.…”