Discrete derived categories I: homomorphisms, autoequivalences and t-structures

Abstract. We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let (U , V) be a t-structure on the bounded derived category D b A with heart H. We investigate when the natural embedding H → D b A can be extended to a triangle equivalence D b H → D b A. Our focus of study is the case where A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t-structure is bounded and the aisle U of the t-structure is closed under the Serre functor.

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“…The following result is a corollary of the description of the bounded t-structures on D b (mod Λ) given in [15] and Proposition 5.5.…”

Section: ) the Object E Is A Silting Object If And Only If (U V) Ismentioning

confidence: 70%

“…It was shown in [15] that the class of derived discrete algebras (introduced in [53]) satisfies this property. We will give a short account in §5.2.…”

Section: ≥0mentioning

confidence: 97%

Derived equivalences for hereditary Artin algebras

Stanley

Roosmalen

2016

Advances in Mathematics

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“…For the claim, we recall from [15, Theorem 2.1] and [4, Proposition 2.3] that A is either derived equivalent to the path algebra of a Dynkin quiver, or to the algebra A(r, N, m) given by the following quiver (1) Recall from [10,5] that the group of standard autoequivalences on D b (A-mod) is explicitly calculated, where A is a derived-discrete algebra of finite global dimension. Theorem 3.6 implies that this group coincides with the whole group of (triangle) autoequivalences on D b (A-mod).…”

Section: Consequently Any Derived Equivalencementioning

confidence: 99%

The derived-discrete algebras and standard equivalences

Chen

Zhang

2019

Journal of Algebra

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“…Example A bounded

t

‐structure is algebraic provided that its heart has finitely many isomorphism classes of indecomposable objects. For example, all bounded

t

‐structures on

{sans-serifD}^{normalb} false(sans-serifmod normalΛ false)

are algebraic for the following finite‐dimensional

K

‐algebra

normalΛ

: (1)

normalΛ

is representation‐finite hereditary; (2)

K

is algebraically closed and

normalΛ

is a finite‐dimensional derived‐discrete

K

‐algebra of finite global dimension [, Proposition 7.1]. …”

Section: Silting Objects and T‐structuresmentioning

confidence: 99%

Discreteness of silting objects and t-structures in triangulated categories

Adachi

Mizuno

Yang

2018

Proc. London Math. Soc.

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We introduce the notion of ST‐pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite‐dimensional algebra. For an ST‐pair (C,D), we construct an injective order‐preserving map from silting objects in sans-serifC to bounded t‐structures on sans-serifD and show that the map is bijective if and only if sans-serifC is silting‐discrete if and only if sans-serifD is t‐discrete. Based on the work of Qiu and Woolf, the above result is applied to show that if sans-serifC is silting‐discrete then the stability space of sans-serifD is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi–Yau triangulated categories associated to Dynkin quivers.

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Discrete derived categories I: homomorphisms, autoequivalences and t-structures

Abstract: Discrete derived categories were studied initially by Vossieck [42] and later by Bobiński, Geiß, Skowroński [9]. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.

Cited by 39 publications

References 57 publications

Derived equivalences for hereditary Artin algebras

Derived equivalences for hereditary Artin algebras

The derived-discrete algebras and standard equivalences

Discreteness of silting objects and t-structures in triangulated categories

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