On global \(\mathscr{C}\)-dimensions
type: Item
instance of: scholarly article
title: On global \(\mathscr{C}\)-dimensions
zbMATH Open document ID: 1431.16002
DOI: 10.1216/RMJ-2019-49-2-557
author: Weiqing Li
author: L. Yan
author: Baiyu Ouyang
published in: Rocky Mountain Journal of Mathematics
publication date: 2019-07-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1561318393
review text: Assume that $R$ is a ring. Let $\mathscr{C}$ be a class of left $R$-modules and $M$ be a left $R$-module. A homomorphism $\phi: C\rightarrow M$ with $C\in\mathscr{C}$ is called $\mathscr{C}$-precover of $M$ if, for any homomorphism $f:C'\rightarrow M$ with $C'\in \mathscr{C}$, there exists a homomorphism $g:C'\rightarrow C$ such that $\phi g=f$. Moreover, if the only such $g$ are automorphisms of $C$ when $C'=C$ and $f=\phi$, then the $\mathscr{C}$-precover $\phi$ is called a $\mathscr{C}$-cover. Dually, a $\mathscr{C}$-preenvelope and a $\mathscr{C}$-envelope can be defined. We say that $\mathscr{C}$ is a (pre)covering (resp., (pre)enveloping) if every left $R$-module has a $\mathscr{C}$-(pre) cover (resp., $\mathscr{C}$-(pre)envelope). Let $\mathscr{C}$ be a class of left $R$-modules and $M$ a left $R$-module. A left $\mathscr{C}$-resolution of $M$ means a $\text{Hom}(\mathscr{C}, -)$ exact complex $\cdots\rightarrow C_1\rightarrow C_0\rightarrow M\rightarrow 0$ (not necessarily exact) with each $C_i\in\mathscr{C}$. Similarly, a right $\mathscr{C}$-resolution of $M$ means a $\text{Hom}(-, \mathscr{C})$ exact complex $0\rightarrow M\rightarrow C^0\rightarrow C^1\rightarrow\cdots $(not necessarily exact) with each $C^i\in\mathscr{C}$. Assume that $\mathscr{C}$ is a precovering. The left $\mathscr{C}$-dimension of $M$, left $\mathscr{C}$-$\dim M$, is defined to be the least positive integer $m$ such that there exists a left $\mathscr{C}$-resolution $0\rightarrow C_m\rightarrow\dots\rightarrow C_1\rightarrow C_0\rightarrow M\rightarrow 0$ of $M$. If there is no such $m$, we say left $\mathscr{C}$-$\dim M=\infty$. The global left $\mathscr{C}$-dimension of $\mathcal{M}$ (where $\mathcal{M}$ is the category of left $R$-modules), $\text{gl left }\mathscr{C}$-$\dim \mathcal{M}$ is defined as $\sup\{\text{left }\mathscr{C}$-$\dim M\mid M\in\mathcal{M}\}$ and is infinite otherwise. The right $\mathscr{C}$-$\dim M$ and the global right $\mathscr{C}$-dimension of $\mathcal{M}…
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 18G25
zbMATH DE Number: 7079984
zbMATH Keywords: global \(\mathscr{C}\)-dimension
zbMATH Keywords: envelope
zbMATH Keywords: cover
reviewed by: Jae Keol Park
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7079984