Tomasz Maszczyk, Andrzej Weber
Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_λ$ and $L_λ$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.
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We show that certain categories of perverse sheaves on a pair of affine toric varieties defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel. The functor expressing this duality is constructed explicitly using a combinatorial model for mixed sheaves on toric varieties.
A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical.
The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah…
Koszul Duality for Symmetric Algebras and Derived Loop Spaces
本文证明了连通复形对称代数的Koszul对偶，并构造了任意有限型分离概形的导出环路空间与切复形对称代数之间的导出范畴等价，推广了Kapranov的光滑情形结果。
Derived category of equivariant coherent sheaves on a smooth toric variety and Koszul duality
本文通过定义光滑环面簇扇上的分次代数层，建立组合Koszul对偶等价，并用模层描述等变凝聚层范畴，最终将完备情形下的对偶解释为Serre函子。
Categorified Koszul duality of algebras
本文提出了一种代数范畴化的科茨尔对偶理论，研究了单向量范畴中的对偶现象，并建立了与阿廷代数相关的稳定单向量无穷范畴的对偶结果。
We prove a duality for factorization homology which generalizes both usual Poincaré duality for manifolds and Koszul duality for $\mathcal{E}_n$-algebras. The duality has application to the Hochschild homology of associative algebras and enveloping algebras of Lie algebras. We interpret our result at the level of topological quantum field theory.
This is a copy of the article by the same authors published in Duke Math. J. (1994).
We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in previous papers of the authors to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.
We obtain a linear algebra data presentation of the category of constructible with respect to perverse triangulation sheaves on a finite simplicial complex. We also establish Koszul duality between the above mentioned category and the category of perverse sheaves constructible with respect to the triangulation
We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.
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.
Koszul Duality for Quasi-split Real Groups
该研究构建了拟分裂实群的范畴化Vogan对偶，证明了Soergel猜想在拟分裂情形下的正确性。
Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with gr U = A is Koszul dual to differential graded algebra (A!,d). We relate the module categories of this dual pair by a tensor-Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg and Soergel.
Let $X$ be a smooth projective Calabi-Yau variety and $L$ a Koszul line bundle on $X$. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring $A$ of $X$ there are formulas similar to the formulas for cohomology number. This similarity is realized via the box-product resolution of the diagonal $Δ_X \subset X \times X$.
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.
The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul complex established by author earlier. Whence, the dependence of Koszul complexes and dependence of dual Koszul complexes follow when one system is linearly expressed through other system. Obtaned results are used in the proof of homotopic equivalence, formulated earlier by the author, of the Koszul complex and dual Koszul complex of a system of non-homogeneous p…
A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobe…
It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.
Let $X$ be a finite connected simplicial complex, and let $δ$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($δ$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,δ)$ and $B(X,δ)$ respectively, and we prove that $A(X,δ)$ and $B(X,δ)$ are Koszul dual to each other. We define the $δ$-perverse topo…
Let $A$ be a Koszul Calabi-Yau algebra. We show that there exists an isomorphism of Batalin-Vilkovisky algebras between the Hochschild cohomology ring of $A$ and that of its Koszul dual algebra $A^!$. This confirms (a generalization of) a conjecture of R.~Rouquier.
Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.
This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.
Derived deformation functors, Koszul duality, and Maurer-Cartan spaces
本文总结了Hinich的导出Maurer-Cartan函子如何建立微分分次李代数与Artinian微分分次交换代数上的导出Schlessinger函子间的等价关系，并探讨了形变问题及更一般Koszul对偶operad对的类似物。
In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved coalgebras and their coBar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras …
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the…
We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these…
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In …
In their previous work, S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let $A$ be an associative algebra and $V$ an $A$-coring whose right algebra $R$ is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra $B$ and a $B$-coring $W$ whose right algebra is the Ringel dual of $R$. We apply our results in small examples to obtain restrictions on the $A_\infty$-structure of the $\textrm{Ext}$-algebra of standard module…
Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials
本文在假设卢斯滕堡猜想成立的条件下，推导了模范畴O中延伸群维数的公式，并利用环面等变科兹尔双模理论，通过周期性卡兹德-卢斯滕堡多项式系数来表达结果。
Koszul Duality for Coherent Sheaves
研究为无限维 Koszul 代数建立 Koszul 对偶性，应用于二次单项式代数等两类代数，还应用于代数几何，得到交换诺特 Koszul 代数定义的射影概型的有界导出范畴的 Koszul 对偶描述及 BGG 型对应。
Universal Koszul Duality for Kac-Moody Groups
本文通过Soergel理论描述，证明了Kac-Moody旗 manifold上真equivariant K-动机与Langlands对偶的constructible monodromic sheaves之间的单调等价，扩展了有限维旗 manifold上的结果。
Si Li and author suggested in that, in some cases, the AdS/CFT correspondence can be formulated in terms of the algebraic operation of Koszul duality. In this paper this suggestion is checked explicitly for $M2$ branes in an $Ω$-background. The algebra of supersymmetric operators on a stack of $K$ $M2$ branes is shown to be Koszul dual, in large $K$, to the algebra of supersymmetric operators of $11$-dimensional supergravity in an $Ω$-background (using the formulation of supergravity in an $Ω$-background presented in arXiv:1610.04144). The twisted form of supergravity that is used here can be …
Relative tensor products and Koszul duality in monoidal oo-categories
本文研究了单向oo-范畴中相对张量积与Koszul对偶的理论，提出了新的双模作用结构并推广了Koszul对偶的应用。
Tilting sheaves for real groups and Koszul duality
本文发展了实解析簇上的倾斜层理论，构造了其到Soergel双模范畴实类似的完全忠实嵌入，并以此给出了Bezrukavnikov和Vilonen定理的纯几何证明。
Hochschild cohomology of the second kind: Koszul duality and Morita invariance
本文研究了第二种Hochschild上同调的定义、性质及其在Koszul对偶和Morita等价下的不变性，展示了其在不同范畴中的应用。
Global Koszul duality
本文通过构建弯曲余代数和代数的模型结构，实现了全局非交换模空间的构造，揭示了dg Koszul对偶的推广形式。
Higher Koszul duality and $n$-affineness
本文在拓扑背景下研究$\mathbb{E}_n$-Koszul对偶性，将其表述为$(\infty,n)$-范畴的等价，并利用余谱定义的余仿射堆刻画对偶性，同时完整刻画了Betti堆的$n$-仿射性。
Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties
本文研究仿生辛簇上DQ代数层的模范畴，证明几何范畴$\u1d4aa$中存在局部生成元，其dg自同态环与所有DQ-模范畴导出等价，推广了Koszul对偶的经典结果。
Stable sheaf cohomology and Koszul--Ringel duality
本文建立稳定层上同调与Koszul--Ringel对偶的联系，证实周期猜想，发现钩子划分Schur函子$\Ext$群的不变性，并给出非零稳定上同调度数的锐界。
Relative Langlands duality and Koszul duality
本文研究了S-对偶的超球面变种在等变量子化下的对偶性，通过Borel子群的equivariant范畴等价，揭示了极化条件下的模范畴与单子范畴之间的关系。
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