arXiv subjects
Publications and source records attributed to Rafael Parra.
We revisit the concepts of $n$-absolutely pure and $n$-flat modules, and their relation with $n$-coherent rings in the sense of Lee. We provide new proofs for some known claims about these modules and rings, and show other new properties. We also give some corrections to the existing literature.
We introduce a unified approach to finiteness conditions in homological algebra and algebraic $\mathsf{K}$-theory by studying the class of $(n,d)$-coherent rings, defined for $n\in\mathbb{N}^*$ and $d\in\mathbb{N}^*\cup \{\infty\}$. This framework simultaneously controls higher finiteness conditions and bounds on the projective dimensions of modules. In the context of algebraic $\mathsf{K}$-theory, we prove that, for a left $(n,d)$-coherent ring, the inclusion $\mathsf{proj}(R)\hookrightarrow\mathsf{FP}_n^{\leq d}(R)$ induces isomorphisms on all nonnegative $\mathsf{K}$-groups. We then introduce the corresponding relative notions of $\mathsf{FP}_n^{\leq d}$-injective, $\mathsf{FP}_n^{\leq d}$-projective, $\mathsf{FP}_n^{\leq d}$-flat, and $\mathsf{FP}_n^{\leq d}$-cotorsion modules, and obtain characterizations of $(n,d)$-coherent rings in terms of these classes. When $d\geq\gD(R)$ or $d=\infty$, our notions recover several previously studied classes and results.
The purpose of this work is to investigate various notions of regularity from the perspective of finiteness conditions, with the ultimate goal of identifying broad classes of rings that are $\mathsf{K}_0$-regular. In this direction, we revisit the classical concepts of coherence and von Neumann regularity, and establish new characterizations. We then focus on the study of \emph{$n$-coherent regular rings}, recently introduced in [31], and analyze their $\mathsf{K}$-theoretic behavior. Finally, we present applications illustrating how these approaches provide examples of $\mathsf{K}_0$-regular rings.
Let $R$ be a ring. In \cite{MD4} Mao and Ding defined an special class of $R$-modules that they called \( FP_n \)-projective $R$-modules. In this paper, we give some new characterizations of \( FP_n \)-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the \( FP_n \)-injective global dimension in terms of \( FP_n \)-projective $R$-modules are obtained. Using the \( FP_n \)-projective dimension of an $R$-module defined by Ouyang, Duan and Li in \cite{Ouy} we introduce a slightly different \( FP_n \)-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of \cite{Ouy} when the ring in question is strong $n$-coherent.
In this paper we give an explicit and pure combinatorial description of the $m$-coloured quivers that appears in the $m$-coloured mutation class of a quiver of type $\mathbb{A}_n$. The $m$-coloured mutation defined by Buan and Thomas in \cite{BT} generalizes the well-known quiver mutation of Fomin and Zelevinsky \cite{FZ}. In particular, our description generalizes a result of Buan and Vatne, \cite{BV}, which we recover when $m=1$.
Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over $RH$, we show that the skew group rings $RG$ and $RH$ share some properties such as being $n$-Gorenstein, $n$-perfect, $n$-coherent, $(n,d)$, Ding-Chen or IF-rings.
We establish connections between the concepts of Noetherian, regular coherent, and regular n-coherent categories for Z-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative K-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the K-theory of Z-linear categories.
Let $R$ be a strong $n$-coherent ring such that each finitely $n$-presented $R$-module has finite projective dimension. We consider $\mathcal{FP}_{n}(R)$ the full subcategory of $R$-Mod of finitely $n$-presented modules. We prove that $\mathcal{FP}_{n}(R)$ is an exact category, $K_{i}(R) = K_{i}(\mathcal{FP}_{n}(R))$ for every $i\geq 0$ and obtain an expression of $\operatorname{Nil}_{i}(R)$.
Given a significative class $F$ of commutative rings, we study the precise conditions under which a commutative ring $R$ has an $F$-envelope. A full answer is obtained when $F$ is the class of fields, semisimple commutative rings or integral domains. When $F$ is the class of Noetherian rings, we give a full answer when the Krull dimension of $R$ is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.