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Vanishing Theorems for Complete Intersections

✍ Scribed by Craig Huneke; David A Jorgensen; Roger Wiegand

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
135 KB
Volume
238
Category
Article
ISSN
0021-8693

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✦ Synopsis

The main focus of his paper was to understand when the tensor product M m N of finitely generated modules M and N over a regular local ring R R is torsion-free. This condition forces the vanishing of a certain Tor module associated with M and N, which in turn, by Auslander's famous R Ε½ . rigidity theorem, implies that Tor M, N s 0 for all i G 1. The vanishing i of Tor carries a great deal of information; for example, it implies the Ε½ . Ε½ . Ε½ . Ε½ . ''depth formula'' depth M q depth N s dim R q depth M m N .

R

From this formula one can deduce, for example, the highly nontrivial fact that if M m N is torsion-free and nonzero, then M and N must both be R torsion-free.

w x Huneke and Wiegand 6 generalized some of Auslander's results to hypersurfaces, though with some unavoidable extra hypotheses. In particuw Ž .x lar, they proved a rigidity theorem 6, 2.4 and the following vanishing theorem: w Ž .x 0.1. THEOREM 6, 2.7 . Let R be a hypersurface, and let M and N be nonzero finitely generated R-modules such that M m N is reflexi¨e. Assume, R

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