scholarly journals Purity and flatness in symmetric monoidal closed exact categories

2019 ◽  
Vol 19 (01) ◽  
pp. 2050004
Author(s):  
E. Hosseini ◽  
A. Zaghian
Keyword(s):  
Abelian Category ◽  
Exact Category ◽  
Injective Cogenerator ◽  
Exact Categories

Let [Formula: see text] be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When [Formula: see text] is endowed with an injective cogenerator with respect to the exact structure, we show that an object [Formula: see text] in [Formula: see text] is flat if and only if any conflation ending in [Formula: see text] is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in [Formula: see text]. In the case [Formula: see text] is a quasi-abelian category, we prove that [Formula: see text] has enough pure injective objects.

scholarly journals The free exact category on a left exact one

1982 ◽  
Vol 33 (3) ◽  
pp. 295-301 ◽  
Author(s):  
A. Carboni ◽  
R. Celia Magno
Keyword(s):  
Abelian Category ◽  
Exact Category ◽  
One Step

AbstractWe give an explicit one step description of the free (Barr) exact category on a left exact one. As an application we give a new two step construction of the free abelian category on an additive one.

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

scholarly journals LOCALLY COMPACT OBJECTS IN EXACT CATEGORIES

2011 ◽  
Vol 22 (12) ◽  
pp. 1787-1821 ◽  
Author(s):  
LUIGI PREVIDI
Keyword(s):  
Compact Objects ◽  
Locally Compact ◽  
Exact Category ◽  
Mutual Relations ◽  
Exact Categories

We identify two categories of locally compact objects on an exact category [Formula: see text]. They correspond to the well-known constructions of the Beilinson category [Formula: see text] and the Kato category [Formula: see text]. We study their mutual relations and compare the two constructions. We prove that [Formula: see text] is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants, and study the exact structure of the category [Formula: see text] of Tate spaces. It is natural therefore to consider the Beilinson category [Formula: see text] as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories [Formula: see text], [Formula: see text] of countably indexed ind/pro-objects over any category [Formula: see text] can be described as localizations of categories of diagrams over [Formula: see text].

STRATIFYING SYSTEMS FOR EXACT CATEGORIES

2018 ◽  
Vol 61 (03) ◽  
pp. 501-521
Author(s):  
VALENTE SANTIAGO
Keyword(s):  
Homological Algebra ◽  
Derived Category ◽  
Classical Sense ◽  
Exact Category ◽  
Relative Homological Algebra ◽  
Stratified Algebra ◽  
Stratifying System ◽  
Exact Categories

AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

scholarly journals Relatively divisible and relatively flat objects in exact categories

2020 ◽  
pp. 2150088
Author(s):  
Septimiu Crivei ◽  
Derya Keski̇n Tütüncü
Keyword(s):  
Approximation Theory ◽  
Closure Properties ◽  
Cotorsion Pair ◽  
Exact Category ◽  
Galois Connections ◽  
Cotorsion Pairs ◽  
Exact Categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

Lattices Induced by Modular Pull-back Exact Categories

2012 ◽  
Vol 19 (04) ◽  
pp. 713-726
Author(s):  
Demei Li ◽  
Lin Xin
Keyword(s):  
Modular Lattices ◽  
Exact Category ◽  
Exact Sequences ◽  
Exact Categories

In this paper, we introduce the notion of a modular pull-back exact category, study modular lattices [Formula: see text] and [Formula: see text] on the skeletally small modular pull-back exact category (𝒞,ℰ) and show that there is an isomorphism between these two lattices. We also study short exact sequences of lattices induced by ℰ-exact sequences.

K1 of Exact Categories by Mirror Image Sequences

2012 ◽  
Vol 11 (1) ◽  
pp. 155-181 ◽  
Author(s):  
Clayton Sherman
Keyword(s):  
Image Sequence ◽  
Image Sequences ◽  
Mirror Image ◽  
Exact Category ◽  
Exact Sequences ◽  
Exact Categories

AbstractWe establish a presentation for K1 of any small exact category P, based on the notion of “mirror image sequence,” originally introduced by Grayson in 1979; as part of the proof, we show that every element of K1(P) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”

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.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

scholarly journals Nori Motives of Curves With Modulus and Laumon 1-motives

2018 ◽  
Vol 70 (4) ◽  
pp. 868-897 ◽  
Author(s):  
Florian Ivorra ◽  
Takao Yamazaki
Keyword(s):  
Number Field ◽  
Abelian Category ◽  
Quiver Representation ◽  
Universal Category ◽  
Effective Divisors

AbstractLet k be a number field. We describe the category of Laumon 1-isomotives over k as the universal category in the sense of M. Nori associated with a quiver representation built out of smooth proper k-curves with two disjoint effective divisors and a notion of for such “curves with modulus”. This result extends and relies on a theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

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