INDECOMPOSABLE CANONICAL MODULES AND CONNECTEDNESS

By M. Hochster, Craig Huneke. 2002

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INDECOMPOSABLE CANONICAL MODULESAND CONNECTEDNESSMelvin Hochster and Craig Huneke1. IntroductionThroughout this paper all rings are commutative, with identity, and Noetherian, unlessotherwise speciﬁed. In particular, “local ring” always means Noetherian local ring, unlessotherwise speciﬁed. Our objective is to prove a generalization of Faltings’ connectednesstheorem [Fal1, Fal2], which asserts that in a complete local domain (R,m,K) of dimensionn, if I ⊆ m is an ideal generated by at most n−2 elements, then the punctured spectrum ofR/I is connected. Our result (see Theorems 3.3and 3.6)draws the same conclusion withoutthe hypothesis that R be a domain: we assume instead that R is complete, equidimensional(i.e., for every minimal prime p of R, dimR/p = dimR), and that one of the followingconditions, which we shall prove are equivalent, holds:a) H

Subfields: Algebra and Number Theory, Geometry and Topology

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Citations in the corpus

On a non-vanishing conjecture of Kawamata and the core of an ideal (2003, background): “The basic properties of canonical modules over (non-Cohen-Macaulay) local rings are summarized, with references, in [ 20, Section 2 ].”
Cohomological dimension of certain algebraic varieties (2002, methodology): “As a consequence of Theorem 1.1(ii), we remove the indecomposability condition in [ 10, Theorem 3.3 ] and [5, Theorem 4.3].”