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Search: MSC category 18G25 ( Relative homological algebra, projective classes )

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1. CMB Online first

Zheng, Yuefei; Huang, Zhaoyong
Triangulated Equivalences Involving Gorenstein Projective Modules
For any ring $R$, we show that, in the bounded derived category $D^{b}(\operatorname{Mod} R)$ of left $R$-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ (resp. $\overline{\mathbf{GI}}(\operatorname{Mod} R)$) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ and $\overline{\mathbf{GI}}(\operatorname{Mod} R)$ are equivalent.

Keywords:triangulated equivalence, Gorenstein projective module, stable category, derived category, homotopy category
Categories:18G25, 16E35

2. CMB 2015 (vol 58 pp. 824)

Luo, Xiu-Hua
Exact Morphism Category and Gorenstein-projective Representations
Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated by all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. The category of all finite-dimensional representations of $(Q, J^2)$ over $A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a subcategory of $\operatorname{rep}{}(Q,J^2,A)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\operatorname{rep}{}(Q,J^2,A)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra, if and only if the Gorenstein-projective representations are exactly the exact representations of $(Q, J^2)$ over $A$.

Keywords:representations of a quiver over an algebra, exact representations, Gorenstein-projective modules

3. CMB 2013 (vol 57 pp. 318)

Huang, Zhaoyong
Duality of Preenvelopes and Pure Injective Modules
Let $R$ be an arbitrary ring and $(-)^+=\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.

Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modules
Categories:18G25, 16E30

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