scholarly journals Parametrizing torsion pairs in derived categories

2021 ◽  
Vol 25 (23) ◽  
pp. 679-731
Author(s):  
Lidia Angeleri Hügel ◽  
Michal Hrbek
Keyword(s):  
Natural Extension ◽  
Commutative Rings ◽  
Derived Categories ◽  
Derived Category ◽  
Content Type ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Zariski Spectrum ◽  
Compactly Generated

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A , which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) . This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.

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 Torsion Pairs and Ringel Duality for Schur Algebras

2021 ◽  
Author(s):  
Karin Erdmann ◽  
Stacey Law
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Schur Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Functorial Approach

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

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 ONE POSITIVE AND TWO NEGATIVE RESULTS FOR DERIVED CATEGORIES OF ALGEBRAIC STACKS

2018 ◽  
Vol 18 (5) ◽  
pp. 1087-1111 ◽  
Author(s):  
Jack Hall ◽  
Amnon Neeman ◽  
David Rydh
Keyword(s):  
Positive Characteristic ◽  
Derived Categories ◽  
Derived Category ◽  
Algebraic Stacks ◽  
Negative Results ◽  
Perfect Complexes ◽  
Compactly Generated

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\text{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $\unicode[STIX]{x1D6F9}_{X}:\mathsf{D}(\mathsf{QCoh}(X))\rightarrow \mathsf{D}_{\text{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.

scholarly journals Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface

2019 ◽  
Author(s):  
Aslak Bakke Buan ◽  
Yu Qiu ◽  
Yu Zhou
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Stability Conditions ◽  
Finite Dimensional ◽  
Quiver With Potential ◽  
Dg Algebra ◽  
Marked Surface

Abstract We study the Ginzburg dg algebra $\Gamma _{\mathbf {T}}$ associated with the quiver with potential arising from a triangulation $\mathbf {T}$ of a decorated marked surface ${\mathbf {S}}_\bigtriangleup$, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories $\operatorname {\mathcal {D}}_{fd}(\Gamma _{\mathbf {T}})$, denoted by $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$. As an application, we show that the spherical twist group $\operatorname {ST}({\mathbf {S}}_\bigtriangleup )$ associated with $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$ acts faithfully on its space of stability conditions.

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.

Piecewise Hereditary Triangular Matrix Algebras

2021 ◽  
Vol 28 (01) ◽  
pp. 143-154
Author(s):  
Yiyu Li ◽  
Ming Lu
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Derived Categories ◽  
Matrix Algebras ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Abelian Categories ◽  
Upper Triangular Matrix ◽  
Orbit Categories ◽  
Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

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