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$𝔸^{1}$ -homotopy theory.

Voevodsky, Vladimir (1998)

Documenta Mathematica

$\infty$ -groupoids and homotopy types

M. M. Kapranov, V. A. Voevodsky (1991)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

$(n, d)$ -injective covers, $n$ -coherent rings, and $(n, d)$ -rings

Weiqing Li, Baiyu Ouyang (2014)

Czechoslovak Mathematical Journal

It is known that a ring $R$ is left Noetherian if and only if every left $R$ -module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$ -coherent ring, then every right $R$ -module has an $(n, d)$ -injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n, 0)$ -injective right $R$ -module is $n$ -pure extending, and if every right $R$ -module has an $(n, 0)$ -injective cover, then $R$ is right $n$ -coherent. As applications of these results, we give some characterizations of $(n, d)$ -rings, von Neumann regular rings and semisimple rings....

$𝒦$ -purity and orthogonality.

Hébert, Michel (2004)

Theory and Applications of Categories [electronic only]

(Co)homology of crossed modules with coefficients in a $π_{1}$ -module.

Paoli, Simona (2003)

Homology, Homotopy and Applications

(Co)homology of triassociative algebras.

Yau, Donald (2006)

International Journal of Mathematics and Mathematical Sciences

(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 $Λ$ .