28 pages, uses xy-pic; v.2 incorporates minor changes throughout, and discussions of semidualizing objects over non-local rings; v. 3 fixes errors in Theorem C and Theorem 4.2; final version to appear in Proceedings for Fifth International Fez Conference on Commutative Algebra and Applications
Let R be a commutative local noetherian ring. We prove that the existence of a chain of semidualizing R-complexes of length (d+1) yields a degree-d polynomial lower bound for the Bass numbers of R. We also show how information about certain Bass numbers of R provide restrictions on the lengths of chains of semidualizing R-complexes. To make this article somewhat self-contained, we also include a survey of some of the basic properties of semidualizing modules, semidualizing complexes and derived categories.
It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=\sup C$ and $d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if…
Let $(R, \mathfrak{m}, k)$ be a commutative Noetherian local ring. We study the suitable chains of semidualizing $R$-modules. We prove that when $R$ is Artinian, the existence of a suitable chain of semidualizing modules of length $n=\mathrm{max}\,\{\,i\geqslant 0\ |\ \mathfrak{m}^{i}\neq 0\,\}$ implies that the the Poincar$\acute{\mathrm{e}}$ series of $k$ and the Bass series of $R$ have very specific forms. Also, in this case we show that the Bass numbers of $R$ are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of $R$.
We prove that a local ring $R$ of embedding codepth at most 3 has at most two semidualizing complexes up to shift-isomorphism, namely, $R$ itself and a dualizing $R$-complex if one exists.
We introduce and study a class of objects that encompasses Christensen and Foxby's semidualizing modules and complexes and Kubik's quasi-dualizing modules: the class of $\mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.
We show that the set $\s(R)$ of shift-isomorphism classes of semidualizing complexes over a local ring $R$ admits a nontrivial metric. We investigate the interplay between the metric and several algebraic operations. Motivated by the dagger duality isometry, we prove the following: If $K,L$ are homologically bounded below and degreewise finite $R$-complexes such that $K\lotimes_R K\lotimes_R L$ is semidualizing, then $K$ is shift-isomorphic to $R$. In investigating the existence of nontrivial open balls in $\s(R)$, we prove that $\s(R)$ contains elements that are not comparable in the reflexiv…
We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_R(B,C) for n>>0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext^n_R(B,C)=0 for all n>>0. We introduce and investigate an equivalence relation \approx on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural act…
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $\mathcal{F}_C$-projective dimension, and finite $\mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
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 no…
We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and C-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new in the commutative, noetherian setting.
We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special conver…
We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings.
Inspired by a recent work of Buchweitz and Flenner, we show that, for a semidualizing bimodule $C$, $C$--perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of $C$--projective modules.
Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing if and only if there exists a Cohen-Macaulay $R$-module of type 1 and of finite G$ _C $-dimension. This result extends Takahashi \cite[Theorem 2.3]{T} as well as Christensen \cite[Proposition 8.4]{C}. Next, as a generalization of Xu \cite[Theorem 3.2]{X2}, we show that $C$ is dualizing if and only if for an $R$-module $M$, the necessary and sufficient conditi…
We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that A^s(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories: \underline{GMor}(R) --> A^s(R) / K^…
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, a number of new results concerning derived reflexivity with respect to C are established. Noteworthy is the statement that derive…
A characterization of Cohen-Macaulay rings in terms of levels of perfect complexes
Institute for the Advancement of Higher Education, Okayama University of Science（冈山理科大学高等教育推进研究所）；Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
