On the invariance and general cohomology comparison theorems

Abstract

The Invariance Theorem of Gerstenhaber and Schack states that if \(\mathbb A \) is a diagram of algebras then the subdivision functor induces a natural isomorphism between the Yoneda cohomologies of the category \(\mathbb A \)-\(\mathbf{mod }\) and its subdivided category \(\mathbb A ^{\prime }\)-\(\mathbf{mod }\). In this paper we generalize this result and show that the subdivision functor is a full and faithful functor between two suitable derived categories of \(\mathbb A \)-\(\mathbf{mod }\) and \(\mathbb A ^{\prime }\)-\(\mathbf{mod }\). This result combined with our work in Stancu (Hochschild cohomology and derived categories, PhD thesis, SUNY, Buffalo, 2006; J Homotopy Relat Struct 6(1):39–63, 2011), on the Special Cohomology Comparison Theorem, constitutes a generalization of Gerstenhaber and Schack’s General Cohomology Comparison Theorem (GCCT).