DG algebra structures on AS-regular algebras of dimension 2

Abstract

Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H(A). Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We (J Algebra, 2008, 320: 2934–2962). Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DG A-module with a free underlying graded A#-module may not be semi-projective.