Dominant dimension and idempotent ideals

Abstract

Let A be a finite dimensional algebra over a field k. In this paper, we study dominant dimension from the point of view of the idempotent ideals. The canonical A-bimodule V:=HomA(DA,A) was studied in [8,11,9], where D=Homk(−,k). Under certain condition, we give a new understanding that DA⊗AV is isomorphic to an idempotent ideal tensor product itself as A-bimodules and show that the double dual functor is defined by DA⊗AV. We also give a characterization of dominant dimension for A in terms of vanishing of certain extension groups over A, which just uses A and DA as ingredients and generalizes the corresponding characterizations given in [8,9]. Moreover, we show that A is a Motita algebra if and only if there is a natural isomorphism ννν−1≅ν on A-mod where ν is the Nakayama functor.