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 FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO

2012 ◽  
Vol 12 (02) ◽  
pp. 1250149 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
DAG OSKAR MADSEN ◽  
XIUPING SU
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Homological Dimension ◽  
Derived Equivalence ◽  
Torsion Pair ◽  
Abelian Categories ◽  
Tilting Object ◽  
Dimension One

We consider filtrations of objects in an abelian category [Formula: see text] induced by a tilting object T of homological dimension at most two. We define three extension closed subcategories [Formula: see text] and [Formula: see text] with [Formula: see text] for j > i, such that each object in [Formula: see text] has a unique filtration with factors in these categories. In dimension one, this filtration coincides with the classical two-step filtration induced by the torsion pair. We also give a refined filtration, using the derived equivalence between the derived categories of [Formula: see text] and the module category of [Formula: see text].

scholarly journals Traits and values as predictors of the frequency of everyday behavior: Comparison between models and levels

2018 ◽  
Vol 40 (1) ◽  
pp. 133-153 ◽  
Author(s):  
Ewa Skimina ◽  
Jan Cieciuch ◽  
Włodzimierz Strus
Keyword(s):  
Personality Traits ◽  
Factor Model ◽  
Five Factor Model ◽  
Personal Values ◽  
Derived Categories ◽  
Higher Order ◽  
Behavioral Factors ◽  
Wide Range ◽  
Age Range ◽  
Hexaco Model

AbstractThe aims of this study were to compare (a) personality traits vs personal values, (b) Five-Factor Model (FFM) vs HEXACO model of personality traits, and (c) broad vs narrow personality constructs in terms of their relationship with the frequency of everyday behaviors. These relationships were analyzed at three organizational levels of self-reported behavior: (a) single behavioral acts, (b) behavioral components (empirically derived categories of similar behaviors), and (c) two higher-order factors. The study was conducted on a Polish sample (N = 532, age range 16–72). We found that (a) even the frequencies of single behavioral acts were related to various personality constructs instead of one narrow trait or value, (b) personality traits and personal values were comparable as predictors of a wide range of everyday behaviors, (c) HEXACO correlated with the frequency of behaviors slightly higher than FFM, and (d) narrow and broad personality constructs did not differ substantially as predictors of everyday behavior at the levels of acts and components, but at the level of higher-order behavioral factors, broad personality measures were better predictors than narrow ones.

Ideal balanced pairs

2020 ◽  
pp. 2150065
Author(s):  
Javad Asadollahi ◽  
Sara Hemat ◽  
Razieh Vahed
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Complete Ideal

In this paper, ideal balanced pairs in an abelian category will be introduced and studied. It is proved that every ideal balanced pair gives rise to a triangle equivalence of relative derived categories. We define complete ideal cotorsion triplets and investigate their relation with ideal balanced pairs.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

scholarly journals The homological projective dual of

2020 ◽  
Vol 156 (3) ◽  
pp. 476-525
Author(s):  
Jørgen Vold Rennemo
Keyword(s):  
Complete Intersection ◽  
Invariant Theory ◽  
Clifford Algebras ◽  
Derived Categories ◽  
Derived Category ◽  
Module Category ◽  
Geometric Invariant ◽  
Projective Duality ◽  
Clifford Module ◽  
Category Of Modules

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

scholarly journals Torsion pairs and filtrations in abelian categories with tilting objects

2015 ◽  
Vol 14 (08) ◽  
pp. 1550121
Author(s):  
Jason Lo
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Explicit Description ◽  
Homological Dimension ◽  
Abelian Categories ◽  
Dimension 2 ◽  
Tilting Object ◽  
Category Of Modules ◽  
Torsion Pairs ◽  
Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

On derived equivalences and homological dimensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Lie Theory ◽  
Restriction Theorem ◽  
Dominant Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Cellular Algebras ◽  
Homological Dimensions ◽  
Finite Dimensional Algebras

AbstractUnlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.

scholarly journals Modified Traces and the Nakayama Functor

2021 ◽  
Author(s):  
Taiki Shibata ◽  
Kenichi Shimizu
Keyword(s):  
Existence And Uniqueness ◽  
Natural Transformation ◽  
Tensor Category ◽  
Abelian Category ◽  
Module Category ◽  
Module Structure ◽  
Exact Functor ◽  
Abelian Categories ◽  
Natural Transformations ◽  
Uniqueness Criteria

AbstractWe organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

Derived Categories: A Quick Tour

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Intermediate Step ◽  
Spectral Sequences ◽  
Separate Section ◽  
Derived Functors ◽  
Left And Right

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.

scholarly journals Derived equivalences and stable equivalences of Morita type, I

2010 ◽  
Vol 200 ◽  
pp. 107-152 ◽  
Author(s):  
Wei Hu ◽  
Changchang Xi
Keyword(s):  
Derived Categories ◽  
Type I ◽  
Sufficient Condition ◽  
Derived Equivalence ◽  
Stable Equivalence ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type ◽  
Finite Dimensional Algebras

AbstractFor self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

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