scholarly journals Representability and autoequivalence groups

2021 ◽  
pp. 1-12
Author(s):  
XIAO–WU CHEN
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Locally Finite ◽  
Finite Dimensional

Abstract For a finite dimensional algebra A, the bounded homotopy category of projective A-modules and the bounded derived category of A-modules are dual to each other via certain categories of locally-finite cohomological functors. We prove that the duality gives rise to a 2-categorical duality between certain strict 2-categories involving bounded homotopy categories and bounded derived categories, respectively. We apply the 2-categorical duality to the study of triangle autoequivalence groups.

On the representation type of subcategories of derived categories

2019 ◽  
Vol 18 (05) ◽  
pp. 2050032
Author(s):  
Chao Zhang
Keyword(s):  
Finite Type ◽  
Type Theorem ◽  
Derived Categories ◽  
Derived Category ◽  
Representation Type ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Length Width ◽  
Contravariantly Finite

Let [Formula: see text] be a finite-dimensional [Formula: see text]-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category [Formula: see text]. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra [Formula: see text], we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories [Formula: see text] of [Formula: see text], that is, [Formula: see text] is of finite type if and only if its cohomological range is finite.

One-point Extensions and Recollements of Derived Categories

2010 ◽  
Vol 17 (03) ◽  
pp. 507-514 ◽  
Author(s):  
Yanan Lin ◽  
Zengqiang Lin
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Arbitrary Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
The One

Let A be a finite dimensional algebra over an arbitrary field k. Assume that a bounded above derived category D-( Mod A) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B and C: [Formula: see text] In this paper, we prove that if there exist M ∈ mod A and N ∈ mod B such that i⋆(N)=M, then the bounded above derived category D-( Mod A[M]) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B[N] and C: [Formula: see text] where A[M] and B[N] are the one-point extensions of A by M and of B by N, respectively. As a consequence, we obtain the main result of Barot and Lenzing [1].

DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

2005 ◽  
Vol 04 (05) ◽  
pp. 587-597 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
XIUPING SU ◽  
ALEXANDER ZIMMERMANN
Keyword(s):  
Partial Order ◽  
Natural Generalization ◽  
Derived Categories ◽  
Derived Category ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.

ON THE NOTION OF DERIVED TAMENESS

2002 ◽  
Vol 01 (02) ◽  
pp. 133-157 ◽  
Author(s):  
CHRISTOF GEISS ◽  
HENNING KRAUSE
Keyword(s):  
Derived Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

The notion of tameness for the derived category of a finite dimensional algebra is introduced and standard properties are established. This is based on classical tameness definitions of Drozd and Crawley-Boevey for the category of finite dimensional representations.

Invariants of finite-dimensional algebras recognized by the semigroup of conjugacy classes of left ideals

2017 ◽  
Vol 16 (10) ◽  
pp. 1750182
Author(s):  
Arkadiusz Mȩcel ◽  
Jan Okniński
Keyword(s):  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Ordinary Quiver

We study the semigroup structure on the set [Formula: see text] of conjugacy classes of left ideals of a finite-dimensional algebra [Formula: see text] over an algebraically closed field [Formula: see text], equipped with the natural multiplication inherited from [Formula: see text], and the structure of the contracted semigroup algebra [Formula: see text]. It is shown that [Formula: see text] has a finite chain of ideals with either nilpotent or completely [Formula: see text]-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver [Formula: see text] of [Formula: see text] is proved to be a subquiver of [Formula: see text], if [Formula: see text] is finite. Moreover, in this case, the structure of [Formula: see text] determines, up to isomorphism, the structure of the algebra [Formula: see text] modulo its Jacobson radical. Combining these results we show that if the semigroup [Formula: see text] is finite, then it determines the structure of any (not necessarily basic) triangular algebra [Formula: see text] which admits a normed presentation.

scholarly journals Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Necessary conditions for power commuting in finite-dimensional algebras over a field

2018 ◽  
Vol 28 (5) ◽  
pp. 339-344
Author(s):  
Andrey V. Zyazin ◽  
Sergey Yu. Katyshev
Keyword(s):  
Necessary Conditions ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Finite Dimensional Algebras

Abstract Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.

scholarly journals Pure derived and pure singularity categories

2019 ◽  
Vol 19 (06) ◽  
pp. 2050117
Author(s):  
Tianya Cao ◽  
Wei Ren
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Projective Modules ◽  
Strong Assumption ◽  
Singularity Category

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective modules, which is called a pure singularity category since we show that it reflects the finiteness of pure-global dimension of rings. Moreover, invariance of pure singularity in a recollement of bounded pure derived categories is studied.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

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