Resolutions of length four which are differential graded algebras. (English) Zbl 1506.13023
Summary: 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 Poincaré 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.
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
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