Examples of Derived Equivalences

2021 ◽  
pp. 329-338
Keyword(s):  
Derived Equivalences

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.

Gorenstein derived equivalences and their invariants

2014 ◽  
Vol 218 (5) ◽  
pp. 888-903 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed
Keyword(s):  
Derived Equivalences

scholarly journals p-Adic lifting problems and derived equivalences

2012 ◽  
Vol 356 (1) ◽  
pp. 90-114 ◽  
Author(s):  
Florian Eisele
Keyword(s):  
Derived Equivalences

scholarly journals Autoequivalences of twisted K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 912-937 ◽  
Author(s):  
Emanuel Reinecke
Keyword(s):  
K3 Surfaces ◽  
K3 Surface ◽  
Lattice Structures ◽  
Hodge Structures ◽  
Derived Equivalences

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$or $2$in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

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