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

TL;DR

This paper explores the structure and size of the set of semidualizing complexes over a local ring, establishing lower bounds on its cardinality based on chain length and properties of ring homomorphisms.

Contribution

It provides new lower bounds for the number of semidualizing complexes, relating chain length and ring homomorphism properties to the size of S(R).

Findings

If S(R) has a chain of length n and reflexivity is transitive, then |S(R)| ≥ 2^n.

The size of S(S) is at least twice that of S(R) when f: R→S is a non-Gorenstein finite flat dimension homomorphism.

Explicit descriptions of the order-structure of S(R) are provided.

Abstract

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).