Lower bounds for the number of semidualizing complexes over a local ring (Q2879610)

Let \((R,\mathfrak{m})\) denote a local noetherian ring. A homologically finite \(R\)-complex \(C\) is called semidualizing, whenever the natural homothethy morphism \(R \to \text{R} \text{Hom}_R(C,C)\) is an isomorphism in the derived category \(\mathcal{D}(R)\). This is a generalization of the notion of a dualizing complex. A semidualizing complex gives rise to a notion of reflexivity for homologically finite \(R\)-complexes. The shift isomorphism classes of semidualizing \(R\)-complexes (ordered via the reflexivity relation) is denoted by \(\mathcal{S}(R)\). This relation is reflexive and antisymmetric. But it is open whether it is transitive in general or whether it is finite.NEWLINENEWLINEIn the paper the author investigates the question whenever the cardinality \(|\mathcal{S}(R)|\) is equal to \(2^n\) for some \(n \in \mathbb{N}\). The main results are: (1) Assume that the reflexivity ordering on \(\mathcal{S}(R)\) is transitive. If \(\mathcal{S}(R)\) admits a chain of length \(n\), then \(|\mathcal{S}(R)| \geq 2^n\). (2) Let \(\phi : R \to S\) denote a local homomorphism of local rings of finite flat dimension. If \(R\) and \(S\) both admit a dualizing complex and \(\phi\) is not Gorenstein, then \(|\mathcal{S}(S)| \geq 2 |\mathcal{S}(R)|\). -- For the statement in (1) there are two proofs. The advantage of the second one is the desribtion of the reflexivity relations in terms of combinatorial data. The result (2) is part of a more general, more technical statement.