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.”

HIGHER -THEORY OF FORMS I. FROM RINGS TO EXACT CATEGORIES

2019 ◽  
pp. 1-69 ◽  
Author(s):  
Marco Schlichting
Keyword(s):  
Form Parameter ◽  
Hermitian Forms ◽  
Group Completion ◽  
Theory Of Forms ◽  
Exact Sequences ◽  
Exact Categories

We prove the analog for the $K$ -theory of forms of the $Q=+$ theorem in algebraic $K$ -theory. That is, we show that the $K$ -theory of forms defined in terms of an $S_{\bullet }$ -construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter.

scholarly journals Ziegler partial morphisms in additive exact categories

2020 ◽  
Vol 10 (03) ◽  
pp. 2050012
Author(s):  
Manuel Cortés-Izurdiaga ◽  
Pedro A. Guil Asensio ◽  
Berke Kalebog̃az ◽  
Ashish K. Srivastava
Keyword(s):  
General Theory ◽  
Category Of Modules ◽  
Exact Sequences ◽  
Exact Categories

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

Double Short Exact Sequences and K1 of an Exact Category

K-Theory ◽  
1998 ◽  
Vol 14 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Alexander Nenashev
Keyword(s):  
Exact Category ◽  
Exact Sequences

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 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 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.

The category of exact sequences of Banach spaces

2010 ◽  
pp. 139-158 ◽  
Author(s):  
Jesús M. F. Castillo ◽  
Yolanda Moreno
Keyword(s):  
Banach Spaces ◽  
Exact Sequences
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