Directed colimits and sheaves in some non-abelian categories

1971 ◽  
pp. 36-69 ◽  
Author(s):  
Pierre Antoine Grillet
Keyword(s):  
Abelian Categories

scholarly journals Configurations in abelian categories—I: Basic properties and moduli stacks

2006 ◽  
Vol 203 (1) ◽  
pp. 194-255 ◽  
Author(s):  
Dominic Joyce
Keyword(s):  
Basic Properties ◽  
Moduli Stacks ◽  
Abelian Categories

scholarly journals Extension of matrix pencil reduction to abelian categories

2018 ◽  
Vol 17 (04) ◽  
pp. 1850062
Author(s):  
Olivier Verdier
Keyword(s):  
Normal Form ◽  
Vector Space ◽  
Decomposition Theorem ◽  
Abelian Category ◽  
Jordan Canonical Form ◽  
Main Technique ◽  
Abelian Categories ◽  
Category Of Modules ◽  
Space Case ◽  
And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Piecewise Hereditary Triangular Matrix Algebras

2021 ◽  
Vol 28 (01) ◽  
pp. 143-154
Author(s):  
Yiyu Li ◽  
Ming Lu
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Derived Categories ◽  
Matrix Algebras ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Abelian Categories ◽  
Upper Triangular Matrix ◽  
Orbit Categories ◽  
Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

Abelian categories

2011 ◽  
pp. 227-231
Author(s):  
J. Adamek ◽  
J. Rosicky ◽  
E. M. Vitale ◽  
F. W. Lawvere
Keyword(s):  
Abelian Categories

Abelian categories

1979 ◽  
pp. 267-323
Author(s):  
Nicolae Popescu ◽  
Liliana Popescu
Keyword(s):  
Abelian Categories

Homology in Relative Semi-Abelian Categories

2012 ◽  
Vol 21 (6) ◽  
pp. 523-543
Author(s):  
Julia Goedecke
Keyword(s):  
Abelian Categories

scholarly journals Divisorial objects in abelian categories

1976 ◽  
Vol 6 (3) ◽  
pp. 539-554
Author(s):  
Shiroh Itoh
Keyword(s):  
Abelian Categories

Abelian Categories

2012 ◽  
pp. 165-183
Author(s):  
Audun Holme
Keyword(s):  
Abelian Categories

Hereditary Abelian Categories and Almost Split Sequences

2001 ◽  
pp. 257-271 ◽  
Author(s):  
Idun Reiten
Keyword(s):  
Abelian Categories ◽  
Almost Split Sequences

Abelian Categories

Algebra ◽  
1973 ◽  
pp. 300-321 ◽  
Author(s):  
Carl Faith
Keyword(s):  
Abelian Categories
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