Second symmetric powers of chain complexes (Q1758773)

The paper ``Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3'' by \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Am. J. Math. 99, 447--485 (1977; Zbl 0373.13006)] launched the study of identifying differential graded algebra structures on finite free resolutions and then used this idea to describe the structure of codimension three Gorenstein ideals. The notion of the the second symmetric power of a complex plays a critical role in the Buchsbaum-Eisenbud paper; although Buchsbaum and Eisenbud implicitly assume that two is a unit in their base ring. The paper under review contains a very careful and thorough account of the second symmetric power of a complex. It carefully states and proves all of the relevant facts from folklore and provides many examples. It compares both versions of the Buchsbaum-Eisenbud construction (with two a unit and the general case) to similar constructions by \textit{A. Dold} and \textit{D. Puppe} [Ann. Inst. Fourier 11, 201--312 (1961; Zbl 0098.36005)] and \textit{A. Tchernev} and \textit{J. Weyman} [J. Algebra 271, No. 1, 22--64 (2004; Zbl 1041.13009)] and it offers a new proof of a Theorem of \textit{L. Avramov, R.-O. Buchweitz} and \textit{L. M. Şega} [J. Pure Appl. Algebra 201, No. 1--3, 218--239 (2005; Zbl 1087.13010)]: the proof involves the second symmetric power of a complex, but the statement does not.