scholarly journals Graded Derived Equivalences

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 103
Author(s):  
Bo-Ye Zhang ◽  
Ji-Wei He
Keyword(s):  
Derived Categories ◽  
Graded Rings ◽  
Graded Algebras ◽  
Derived Equivalences ◽  
Group Gradings ◽  
Morita Type

We consider the equivalences of derived categories of graded rings over different groups. A Morita type equivalence is established between two graded algebras with different group gradings. The results obtained here give a better understanding of the equivalences of derived categories of graded rings.

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.

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

2016 ◽  
Vol 23 (03) ◽  
pp. 385-408
Author(s):  
Xinhong Chen ◽  
Ming Lu
Keyword(s):  
Derived Categories ◽  
Graded Algebras ◽  
Coherent Sheaves ◽  
Derived Equivalences ◽  
Differential Graded Algebras ◽  
Triangular Algebras

In this paper, we first prove for two differential graded algebras (DGAs) A, B which are derived equivalent to k-algebras Λ, Γ, respectively, that [Formula: see text]. In particular, [Formula: see text]. Secondly, for two quasi-compact and separated schemes X, Y and two algebras A, B over k which satisfy [Formula: see text] and [Formula: see text], we show that [Formula: see text] and [Formula: see text]. Finally, we prove that if X is a quasi-compact and separated scheme over k, then [Formula: see text] admits a recollement relative to [Formula: see text], and we describe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

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 equivalence F between the derived categories of Artin algebras A and B arises naturally as a functor between their stable module categories, which can be used to compare certain homological dimensions of A with that of B. We then give a sufficient condition for the functor to 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.

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 On H-simple not necessarily associative algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950162
Author(s):  
A. S. Gordienko
Keyword(s):  
Associative Algebra ◽  
Hopf Algebras ◽  
Simple Algebra ◽  
Polynomial Identities ◽  
Associative Algebras ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Graded Polynomial Identities ◽  
Group Gradings ◽  
Polynomial Formula

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Rigidity dimension of algebras

2019 ◽  
pp. 1-27
Author(s):  
HONGXING CHEN ◽  
MING FANG ◽  
OTTO KERNER ◽  
STEFFEN KOENIG ◽  
KUNIO YAMAGATA
Keyword(s):  
Hochschild Cohomology ◽  
Global Dimension ◽  
Homological Dimension ◽  
Dominant Dimension ◽  
Finite Global Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type

Abstract A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

scholarly journals On semi-infinite cohomology of finite-dimensional graded algebras

2010 ◽  
Vol 146 (2) ◽  
pp. 480-496 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Leonid Positselski
Keyword(s):  
Quantum Group ◽  
General Setting ◽  
Derived Categories ◽  
Graded Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Small Quantum ◽  
Definition Of

AbstractWe describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

scholarly journals Derived Equivalences of Upper Triangular Differential Graded Algebras

2011 ◽  
Vol 39 (7) ◽  
pp. 2367-2387 ◽  
Author(s):  
Daniel Maycock
Keyword(s):  
Graded Algebras ◽  
Derived Equivalences ◽  
Differential Graded Algebras

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 Rank functions on triangulated categories

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Rank Function ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Graded Algebras ◽  
Artinian Rings ◽  
Differential Graded Algebras ◽  
Skew Fields ◽  
Rank Functions

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

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