We are going to show that the sheafication of graded Koszul modules $% K_Γ$ over $Γ_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_Γ$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_Γ$up to shift.
More importantly, the category $K_Γ$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $Γ$ and over the exterior algebra $Λ.$ This is then used to develop a kind of relative Auslander-Reiten theory for the category $\mathit{Coh(P}^{n})$, with respect to this theory, all but finitely many Auslander-Reiten components for $\mathit{Coh(P}^{n})$ have the shape \textit{ZA}$_{\infty}.$ We also describe the remaining ones.
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
Building on the theory of parity sheaves due to Juteau-Mautner-Williamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche-Soergel-Williamson).
For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category O for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.
We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study categories $\operatorname{DK}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show that there is a natural and geometric functor, called Beilinson realisation, from $\mathcal{S}$-constructible mixed sheaves $\operatorname{D}^{mix}_{\mathcal{S}}(X)$ to $\operatorname{DK}_{\m…
We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
We give the full answer to the question: on which curves the category of coherent sheaves $\Coh_{X}$ is tame. The answer is: these are just the curves from the list of Drozd-Greuel. Moreover, in this case the derived category $D^{-}(\Coh_{X})$ is also tame. We give an explicit description of the objects of this category as well as of the categories $D^{b}(\Coh_{X})$, $\Coh_{X}$. Among the coherent sheaves we describe the vector bundles, torsion-free sheaves, mixed sheaves and skyscraper sheaves.
Let ${\bf P}^n$ be the projective $n-$space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on ${\bf P}^n$ has a trivial endomorphism algebra. This generalizes a result of Drezet for $n=2.$
These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus…
Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we classify all large (that is, non-coherent) tilting sheaves and the corresponding resolving classes. In particular we show that in the elliptic and in the tubular cases every large tilting sheaf has a well-defined slope.
In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can classify the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.
In this paper, using the quantum McKay correspondence, we construct the "derived category" of G-equivariant sheaves on the quantum projective line at a root of unity. More precisely, we use the representation theory of U_{q}sl(2) at root of unity to construct an analogue of the symmetric algebra and the structure sheaf. The analogue of the structure sheaf is, in fact, a complex, and moreover it is a dg-algebra. Our derived category arises via a triangulated category of G-equivariant dg-modules for this dg-algebra. We then relate this to representations of the quiver (Γ, \Om), where Γis the A,D…
We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.
We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a…
In this paper we show that if $Λ=\amalg_{i\geq 0}Λ_i$ is a Koszul algebra with $Λ_0$ isomorphic to a product of copies of a field, then the minimal projective resolution of $Λ_0$ as a right $Λ$-module provides all the information necessary to construct both a minimal projective resolution of $Λ_0$ as a left $Λ$-module and a minimal projective resolution of $Λ$ as a right module over the enveloping algebra of $Λ$. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of $Λ_0$ as a right $Λ$-module.
In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori-Matsumoto involution for graded affine Hecke algebras (due to Evens and the first author) and for usual affine Hecke algebras (obtained in a previous paper).
For the base field of complex numbers we discuss the relationship between categories of coherent sheaves on compact Riemann surfaces and categories of coherent sheaves on weighted smooth projective curves. This is done by bringing back to life an old theorem of Bundgaard-Nielsen-Fox proving Fenchel's conjecture for fuchsian groups.
We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category $\mathcal{D}^b(\mathbb{X})$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of virtual genus $\leq 1$. We will see from our description that the combinatorics in classification of bounded t-structures on $\mathcal{D}^b(\mathbb{X})$ can be reduced to that in classification of bounded t-structures on bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras.
We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.
We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.
We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.
Given a finite-dimensional algebra $Λ$ and $A \geqslant 1$, we construct a new algebra $\tildeΛ_A$, called the stretched algebra, and relate the homological properties of $Λ$ and $\tildeΛ_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $Λ$ has (Fg) if and only if $\tildeΛ_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $Λ$ is a $d$-Koszul algebra for some $d \geqslant 2$.
Let $Λ=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $Λ$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $Λ$ over the enveloping algebra $Λ^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $Ł$ relative to the resolution $\mathbb{P}$ is given as a cup product and als…
We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are finite, and develop the tilting theory for such algebras. Under the assumption of existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both Ringel and Koszu…
Let $A = \bigoplus_{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying a certain splitting condition. In this paper we develop a generalized Koszul theory preserving many classical results. Moreover, we define a quotient graded algebra $\bar{A} = \bigoplus_{i \geqslant 0} \bar{A}_i$ and show that $A$ is a generalized Koszul algebra if and only if $\bar{A}$ is a classical Koszul algebra and a projective $A_0$-module. We also describe an application of this theory to the extension algebras of standard modules of standa…
We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D_k constructible with respect to the Schubert stratification. The connection is given by an explicit isomorphism to the endomorphism algebra of a projective generator described in by Braden. The algebra is obtained by a "folding" procedure from the generalized Khovanov arc algebras. We relate this algebra to the category of finite dimensional representations of the …
In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to $p$-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles.
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
We show that the Brauer algebra over the complex numbers for an integral parameter delta can be equipped with a grading, in the case of delta being non-zero turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter N. The parameter delta appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in pa…
