Lower bounds for the number of semidualizing complexes over a local ring

Details Lower bounds for the number of semidualizing complexes over a local ring Lower bounds for the number of semidualizing complexes over a local ring

2008-01-30

arxiv.org/abs/0801.4743v2

Mathematics

Commutative Algebra

v2: title changed, section 4 added, minor changes throughout; 10 pages

Scientific paper

We investigate the set S(R) of shift-isomorphism classes of semidualizing R-complexes, ordered via the reflexivity relation, where R is a commutative noetherian local ring. Specifically, we study the question of whether S(R$ has cardinality 2^n for some n. We show that, if there is a chain of length n in S(R) and if the reflexivity ordering on S(R) is transitive, then S(R) has cardinality at least 2^n, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism f: R\to S of finite flat dimension, if R and S admit dualizing complexes and if f is not Gorenstein, then the cardinality of S(S) is at least twice the cardinality of S(R).

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