scholarly journals Derived Equivalences for the Flops of Type C2 and ${A}_{4}^{G}$ via Mutation of Semiorthogonal Decomposition

2021 ◽  
Author(s):  
Hayato Morimura
Keyword(s):  
Derived Equivalence ◽  
Theor Phys ◽  
Derived Equivalences

AbstractWe give a new proof of the derived equivalence of a pair of varieties connected by the flop of type C2 in the list of Kanemitsu (2018), which is originally due to Segal (Bull. Lond. Math. Soc., 48 (3) 533–538, 2016). We also prove the derived equivalence of a pair of varieties connected by the flop of type ${A}_{4}^{G}$ A 4 G in the same list. The latter proof follows that of the derived equivalence of Calabi–Yau 3-folds in Grassmannians Gr(2,5) and Gr(3,5) by Kapustka and Rampazzo (Commun. Num. Theor. Phys., 13 (4) 725–761 2019) closely.

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.

scholarly journals Rational points and derived equivalence

2021 ◽  
Vol 157 (5) ◽  
pp. 1036-1050
Author(s):  
Nicolas Addington ◽  
Benjamin Antieau ◽  
Katrina Honigs ◽  
Sarah Frei
Keyword(s):  
Rational Point ◽  
Source Code ◽  
Rational Points ◽  
The Other ◽  
Online Version ◽  
Hasse Principle ◽  
Derived Equivalence ◽  
Derived Equivalences ◽  
Closed Fields ◽  
Supplementary Material

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

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 DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES

2018 ◽  
Vol 61 (03) ◽  
pp. 557-573
Author(s):  
SHENGYONG PAN ◽  
ZHEN PENG ◽  
JIE ZHANG
Keyword(s):  
Affirmative Answer ◽  
Derived Equivalence ◽  
Graded Algebras ◽  
Cohomology Rings ◽  
Derived Equivalences ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Differential Graded Algebras ◽  
Endomorphism Algebras ◽  
Special Case

AbstractIn this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

scholarly journals Deformation of Tilting-type Derived Equivalences for Crepant Resolutions

2018 ◽  
Vol 2020 (13) ◽  
pp. 4062-4102
Author(s):  
Wahei Hara
Keyword(s):  
Derived Categories ◽  
Algebraic Varieties ◽  
Derived Equivalence ◽  
Derived Equivalences ◽  
Crepant Resolutions

Abstract We say that an exact equivalence between the derived categories of two algebraic varieties is tilting type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.

scholarly journals Liftable derived equivalences and objective categories

2020 ◽  
Vol 52 (5) ◽  
pp. 816-834
Author(s):  
Xiaofa Chen ◽  
Xiao‐Wu Chen
Keyword(s):  
Derived Equivalences

scholarly journals Perverse Schobers and Wall Crossing

2017 ◽  
Vol 2019 (18) ◽  
pp. 5777-5810 ◽  
Author(s):  
W Donovan
Keyword(s):  
Invariant Theory ◽  
Local System ◽  
Vector Spaces ◽  
Geometric Invariant ◽  
Perverse Sheaf ◽  
Derived Equivalences ◽  
Grothendieck Groups ◽  
Wall Crossing ◽  
Git Quotient ◽  
Cohomology Complex

Abstract For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

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