Gorenstein rings call the tune

Details Gorenstein rings call the tune Gorenstein rings call the tune

2004-07-16

arxiv.org/abs/math/0407289v1

Mathematics

Commutative Algebra

11 pages; to appear in Proceedings of back-to-back workshops at Sevilla/Lisboa in June 2003, Marcel Dekker Lecture Notes

Scientific paper

Hochster's Monomial Conjecture and Canonical Element Conjecture date back some thirty resp. twenty years. They concern all noetherian commutative local rings, and were proved by their originator right away in equal characteristic. Thanks to recent important progress by Heitmann, they are now known to hold in mixed characteristic for all such rings of dimension < 4. Nevertheless, the remaining cases are among the best guarded secrets in this area. We show that these conjectures can be considered as statements about Gorenstein rings of the same dimension and particular characteristic and their module theory. In this survey, we string together relevant results from a series of previous joint articles. No proofs are given, but the ideas behind them are explained. In particular, we propagate the use of Auslander-Buchweitz methods. Thus one obtains a number of different looking results about Gorenstein rings in the known cases. In the remaining ones, it is enough to prove any one of these statements for Gorenstein rings.

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