Second symmetric powers of chain complexes

We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ 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 c{S}ega: let $Rto S$ 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 $cup_nass_R(HH_n(Xotimes_SX))subseteqass(R)$ and if $X_{p}simeq S_{p}$ for each $pinass(R)$, then $Xsimeq S$.