Second symmetric powers of chain complexes

TL;DR

This paper explores the construction of the second symmetric power of chain complexes over rings, providing new results, explicit examples, and a significant application to module-finite ring homomorphisms.

Contribution

It offers new theoretical results, explicit computations, and an application of the second symmetric power construction to a problem in commutative algebra.

Findings

Established several folklore results with new proofs.

Provided explicit computations and examples of S^2_R(X).

Proved a new criterion for complexes to be isomorphic to the ring under certain conditions.

Abstract

We investigate Buchbaum and Eisenbud's construction of the second symmetric power S^2_R(X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and Sega: Let R \to S be a module-finite ring homomorphism such that R is noetherian and local, and such that 2 is a unit in R. Let X be a complex of finite rank free S-modules such that X_n = 0 for each n < 0. If \cup_n Ass_R(H_n(X \otimes_S X)) \subseteq Ass(R) and if X_P \simeq S_P for each P \in Ass(R), then X \simeq S.