Model structures on modules over Ding-Chen rings

By James Gillespie. 2009

149 citations recorded by the source corpus.

An $n$-FC ring is a left and right coherent ring whose left and right self FP-injective dimension is $n$. The work of Ding and Chen in \cite{ding and chen 93} and \cite{ding and chen 96} shows that these rings possess properties which generalize those of $n$-Gorenstein rings. In this paper we call a (left and right) coherent ring with finite (left and right) self FP-injective dimension a Ding-Chen ring. In case the ring is Noetherian these are exactly the Gorenstein rings. We look at classes of modules we call Ding projective, Ding injective and Ding flat which are meant as analogs to Enochs' Gorenstein projective, Gorenstein injective and Gorenstein flat modules. We develop basic properties of these modules. We then show that each of the standard model structures on Mod-$R$, when $R$ is a Gorenstein ring, generalizes to the Ding-Chen case. We show that when $R$ is a commutative Ding-Chen ring and $G$ is a finite group, the group ring $R[G]$ is a Ding-Chen ring.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/454ea8d7aa211d9484198c4de8a3595f2fbe8cf0

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

The stable module category of a general ring (2014, methodology): “ Gorenstein AC-projective modules. This model structure generalizes the Gorenstein projective model structure for Gorenstein rings constructed in [Hov02], as well as its generalization constructed in [Gil10]. The proof of Theorem 8.5 is dual to the proof of Theorem 5.5, and so we leave it to the rea”
Hereditary abelian model categories (2015, background): “As was suggested in [27], these modules are now often called Ding injective, and they turn out to give a quite a satisfactory extension of Gorenstein injective modules to coherent rings.”
On purity and applications to coderived and singularity categories (2014, methodology): “theory of triangulated categories, and explains some phenomena in variants of Kasparov’s bivariant K-theory of C∗-algebras. Another approach was considered by Ding and Chen [DC93, DC96] and Gillespie [Gil10]. They studied two-sided coherent rings with ﬁnite self-fpinjective dimension. This is formal”