  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.
