Resolutions of length four which are Differential Graded Algebras

TL;DR

This paper demonstrates that a length four, self-dual acyclic complex of finitely generated free modules over a commutative Noetherian ring can be endowed with a Differential Graded Algebra structure with Divided Powers, exhibiting Poincaré duality, removing previous restrictions.

Contribution

It extends known results by removing assumptions that the ring is local or Gorenstein and that the complex is minimal, broadening the applicability of the DGA structure.

Findings

F complex admits a DGA with Divided Powers

The multiplication exhibits Poincaré duality

Results hold over general commutative Noetherian rings

Abstract

Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a Differential Graded Algebra with Divided Powers; furthermore, the multiplication on $F$ exhibits Poincar\'e duality. This result is already known if $P$ is a local Gorenstein ring and $F$ is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that $P$ is local, $P$ is Gorenstein, and $F$ is minimal.