Almost Koszul algebras and stably Calabi–Yau algebras

Stably Calabi-Yau properties of derivation quotient algebras

“…We also recall the following result that is Theorem 3.7 in [Yu12]. It is the analogous of [BSW10, Theorem 6.8] in case of Frobenius algebras, establishing a conncetion between almost Koszul Frobenius algebras and stably Calabi-Yau algebras.…”

“…This is the twisted version of Lemma 6.4, [BSW10, Section 6]. A more detailed proof in the case of twisted derivation algebras can be found in [Yu12], in the proofs of Theorem 3.4 and Theorem 3.7.…”

“…This implies that the main results of [Yu12] actually hold for Frobenius almost Koszul algebras, since such algebras are automatically of periodic type.…”

“…In the case of selfinjective algebras, the stably Calabi-Yau property has been introduced and studied for example in [BS07], [Dug12] and [ES06] in the classification of finite and tame representation type algebras. In [Yu12] the author studies the connection between almost Koszul (Frobenius) stably Calabi-Yau algebras and twisted derivation algebras with some exactness conditions on the associated complex. In particular Yu proves that a Frobenius algebra A is (p, q)-Koszul and if and only if it is isomorphic to a twisted derivation algebra such that the associated complex gives the first q terms of a bimodule projective resoltion for A.…”

“…The first goal of this paper is to study twisted Calabi-Yau properties for the stable category of modules over selfinjective algebras and give an interpretation of the results from [Yu12] under this point of view. A preliminary step in this direction is given by the following fact:…”