Let be a commutative Noetherian ring and be a self-dual acyclic complex of finitely generated free -modules. Assume that has length four and has rank one. We prove that can be given the structure of a differential graded algebra with divided powers; furthermore, the multiplication on exhibits Poincaré duality. This result is already known if is a local Gorenstein ring and is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that is local, is Gorenstein, and is minimal.
"Resolutions of length four which are differential graded algebras." J. Commut. Algebra 12 (4) 509 - 538, Winter 2020. https://doi.org/10.1216/jca.2020.12.509