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Let $R$ be a ring, $n$ a fixed non-negative integer, $𝒲 ℐ$ the class of all left $R$ -modules with weak injective dimension at most $n$ , and $𝒲 ℱ$ the class of all right $R$ -modules with weak flat dimension at most $n$ . Using left (right) $𝒲 ℐ$ -resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that $- \otimes -$ is right balanced on $ℳ_{R} \times_{R} ℳ$ by $𝒲 ℱ \times 𝒲 ℐ$ , and investigate the global right $𝒲 ℐ$ -dimension of $_{R} ℳ$ by right derived functors of $\otimes$ .

Some recent results on Hochschild homology of commutative algebras.

Ana Lago, Antonio García Rodicio (1995)

Publicacions Matemàtiques

This article is a brief survey of the recent results obtained by several authors on Hochschild homology of commutative algebras arising from the second author’s paper [21].

Some results on $G_{C}$ -flat dimension of modules

Ramalingam Udhayakumar, Intan Muchtadi-Alamsyah, Chelliah Selvaraj (2019)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we study some properties of $G_{C}$ -flat $R$ -modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_{C}$ -yoke with the $C$ -yoke of a module as well as the relation between the $G_{C}$ -flat resolution and the flat resolution of a module over $G F$ -closed rings. We also obtain a criterion for computing the $G_{C}$ -flat dimension of modules.

Spectral sequences and relative cohomology

Najar, Rudolph M. (1977)

Portugaliae mathematica

Stable short exact sequences and the maximal exact structure of an additive category

Wolfgang Rump (2015)

Fundamenta Mathematicae

It was recently proved that every additive category has a unique maximal exact structure, while it remained open whether the distinguished short exact sequences of this canonical exact structure coincide with the stable short exact sequences. The question is answered by a counterexample which shows that none of the steps to construct the maximal exact structure can be dropped.

Steenrod Squares in Cotor

H. Uehara, B. Al-Hashimi (1974)

Manuscripta mathematica

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Chao Wang, Xiao Yan Yang (2017)

Czechoslovak Mathematical Journal

Let $Λ = (\begin{matrix} A & M \\ 0 & B \end{matrix})$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $Λ$ -modules under the condition that $M$ is a cocompatible $(A, B)$ -bimodule, we establish a recollement of the stable category $\bar{Ginj (Λ)}$ . We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $Λ$ .

Strongly $𝒲$ -Gorenstein modules

Husheng Qiao, Zongyang Xie (2013)

Czechoslovak Mathematical Journal

Let $𝒲$ be a self-orthogonal class of left $R$ -modules. We introduce a class of modules, which is called strongly $𝒲$ -Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $𝒲$ -Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $𝒲$ -Gorenstein module can be inherited by its submodules and quotient modules....

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Relative cohomolgy of groups.

Relative cyclic homology and the Bass conjecture.

Relative derived functors and the homology of groups

Relative EXT groups of Abelian categories

Relative homological categories.

Relative weak derived functors