# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.12405