Hyman Bass and ubiquity: Gorenstein rings. (English) Zbl 0960.13008

Huneke, Craig

Lam, T. Y. (ed.) et al., Algebra, \(K\)-theory, groups, and education. On the occasion of Hyman Bass’s 65th birthday. Mainly the proceedings of the conference, Columbia University, New York, NY, November 6-7, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 243, 55-78 (1999).

The article is an excellent survey of the “ubiquity of Gorenstein rings”, and appreciates very well Hyman Bass’s work and his famous ubiquity paper [H. Bass, Math. Z. 82, 8-28 (1963; Zbl 0112.26604)]. – It starts with plane curves and the state of commutative algebra around 1960, then gives the definition of Gorenstein rings according to Grothendieck and Serre together with significant examples. Next injective modules and Matlis duality are discussed before the “ubiquity” is reached. Bass’s conjecture and its influence on various homological themes are outlined. Further topics are inverse powers and zero-dimensional Gorenstein rings, Hilbert functions, and the behaviour of the Gorenstein property under group operations. A final paragraph is devoted to the module theory over Gorenstein rings.
For the entire collection see [Zbl 0928.00073].

Reviewer: U.Vetter (Oldenburg)

MSC:

13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13-03	History of commutative algebra
01A60	History of mathematics in the 20th century

Keywords:

ubiquity of Gorenstein rings; Matlis duality; Gorenstein property