scholarly journals Koszul duality for compactly generated derived categories of second kind

2021 ◽  
Author(s):  
Ai Guan ◽  
Andrey Lazarev
Keyword(s):  
Derived Categories ◽  
Koszul Duality ◽  
Compactly Generated

scholarly journals Linear Koszul duality

2009 ◽  
Vol 146 (1) ◽  
pp. 233-258 ◽  
Author(s):  
Ivan Mirković ◽  
Simon Riche
Keyword(s):  
Vector Bundle ◽  
Hecke Algebras ◽  
Derived Categories ◽  
Koszul Duality

AbstractIn this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection $F_1 \rcap _E F_2$, and the corresponding derived intersection $F_1^{\perp } \rcap _{E^*} F_2^{\perp }$. We also propose applications to Hecke algebras.

scholarly journals On the abelianization of derived categories and a negative solution to Rosický’s problem

2012 ◽  
Vol 149 (1) ◽  
pp. 125-147 ◽  
Author(s):  
Silvana Bazzoni ◽  
Jan Šťovíček
Keyword(s):  
Regular Cardinal ◽  
Large Family ◽  
Global Dimension ◽  
Derived Categories ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Negative Solution ◽  
Compactly Generated

AbstractWe prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

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.

scholarly journals The symplectic geometry of higher Auslander algebras: Symmetric products of disks

2021 ◽  
Vol 9 ◽  
Author(s):  
Tobias Dyckerhoff ◽  
Gustavo Jasso ◽  
Yankι Lekili
Keyword(s):  
Unit Disk ◽  
Symplectic Geometry ◽  
Morita Equivalence ◽  
Derived Categories ◽  
Symmetric Product ◽  
Koszul Duality ◽  
Fukaya Categories ◽  
Simplicial Space ◽  
Dimensional Unit ◽  
Geometric Realisation

Abstract We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

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 KOSZUL CALCULUS

2017 ◽  
Vol 60 (2) ◽  
pp. 361-399 ◽  
Author(s):  
ROLAND BERGER ◽  
THIERRY LAMBRE ◽  
ANDREA SOLOTAR
Keyword(s):  
Duality Theorem ◽  
Derived Categories ◽  
Quadratic Algebra ◽  
Koszul Duality ◽  
True Nature ◽  
Quadratic Algebras ◽  
Koszul Homology ◽  
Cup Products ◽  
Koszul Cohomology

AbstractWe present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved foranyquadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

scholarly journals Perfect complexes on algebraic stacks

2017 ◽  
Vol 153 (11) ◽  
pp. 2318-2367 ◽  
Author(s):  
Jack Hall ◽  
David Rydh
Keyword(s):  
Derived Categories ◽  
Triangulated Categories ◽  
Azumaya Algebras ◽  
Algebraic Stacks ◽  
Coherent Sheaves ◽  
Algebraic Stack ◽  
Perfect Complexes ◽  
Compactly Generated

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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