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Realization of long exact sequences of abelian groups.

Irwin S. Pressman (1990)

Publicacions Matemàtiques

Given a long exact sequence of abelian groupsL: ... → Li-1 →ξi-1 Li →ξi Li+1 → ...a short exact sequence of complexes of free abelian groups is constructed whose cohomology long exact sequence is precisely L. In this sense, L is realized. Two techniques which are introduced to reduce or replace lengthy diagram chasing arguments may be of interest to some readers. One is an arithmetic of bicartesian squares; the other is the use of the fact that categories of morphisms of abelian categories...

Reedy categories which encode the notion of category actions

Julia E. Bergner, Philip Hackney (2015)

Fundamenta Mathematicae

We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard...

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring and let C be a semidualizing R -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G C -injective module G , the character module G + is G C -flat, then the class 𝒢ℐ C ( R ) ∩ 𝒜 C ( R ) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class 𝒢ℐ C ( R ) ∩ 𝒜 C ( R ) is covering....

Relative weak derived functors

Panneerselvam Prabakaran (2020)

Commentationes Mathematicae Universitatis Carolinae

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 - ⊗ - is right balanced on ℳ R × R ℳ by 𝒲 ℱ × 𝒲 ℐ , and investigate the global right 𝒲 ℐ -dimension of R ℳ by right derived functors of ⊗ .

Salvetti complex, spectral sequences and cohomology of Artin groups

Filippo Callegaro (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

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