Multiplicities and enumeration of semidualizing modules

Details Multiplicities and enumeration of semidualizing modules Multiplicities and enumeration of semidualizing modules

2010-01-15

arxiv.org/abs/1001.2632v1

Mathematics

Commutative Algebra

12 pages, uses xypic

Scientific paper

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel multiplicities e_R(J;C) = e_R(J;R) for all semidualizing R-modules C and all m-primary ideals J. The classes of rings we investigate include those that are determined by ideals defining fat point schemes in projective space or by monomial ideals. We use these ideas to show that if R is local (or graded) and Cohen-Macaulay with codimension 2, then R has at most two (graded) semidualizing modules, up to isomorphism, namely R and a dualizing module.

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