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 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].

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.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

scholarly journals τ-Complexity and Tilting Modules

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Lijing Zheng ◽  
Chonghui Huang ◽  
Qianhong Wan
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Finite Dimensional

Let A be a finite dimensional algebra over an algebraic closed field k. In this note, we will show that if T is a separating and splitting tilting A-module, then τ-complexities of A and B are equal, where B=EndA(T).

Relative Quotient Triangulated Categories

2014 ◽  
Vol 21 (02) ◽  
pp. 195-206 ◽  
Author(s):  
Shengyong Pan
Keyword(s):  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Stable Category ◽  
Gorenstein Algebra ◽  
Finite Dimensional ◽  
Frobenius Category ◽  
Relative Version ◽  
Relative Derived Category ◽  
Algebra Over A Field

Let A be a finite dimensional algebra over a field k. We consider a subfunctor F of [Formula: see text], which has enough projectives and injectives such that [Formula: see text] is of finite type, where [Formula: see text] denotes the set of F-projectives. One can get the relative derived category [Formula: see text] of A-mod. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category [Formula: see text] and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equivalence between [Formula: see text] and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.

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.

scholarly journals -tilting theory

2013 ◽  
Vol 150 (3) ◽  
pp. 415-452 ◽  
Author(s):  
Takahide Adachi ◽  
Osamu Iyama ◽  
Idun Reiten
Keyword(s):  
Direct Summand ◽  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Tilting Theory ◽  
Finite Dimensional ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

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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