A functor of sets $\mathbb X$ over the category of $K$-commutative algebras is said to be an affine functor if its functor of functions, $\mathbb A_{\mathbb X}$, is reflexive and $\mathbb X=\Spec \mathbb A_{\mathbb X}$. We prove that affine functors are equal to a direct limit of affine schemes and that affine schemes, formal schemes, the completion of affine schemes along a closed subscheme, etc., are affine functors.
Endowing an affine functor $\mathbb X$ with a functor of monoids structure is equivalent to endowing $\mathbb A_{\mathbb X}$ with a functor of bialgebras structure. If $\mathbb G$ is an affine functor of monoids, then $\mathbb A_{\mathbb G}^*$ is the enveloping functor of algebras of $\mathbb G$ and the category of $\mathbb G$-modules is equivalent to the category of $\mathbb A_{\mathbb G}^*$-modules. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes, the equivalence between formal groups and Lie algebras in characteristic zero, etc.
We introduce the formal extension of the Whitney functor and the polynomial extension of the tempered cohomology functor, and prove a natural topological duality between them.
In this paper we obtain the Cartier duality for k-schemes of commutative monoids functorially without providing the vector spaces of functions with a topology, generalizing a result for finite commutative algebraic groups by M. Demazure and P. Gabriel.
We extend to the category of relative regular holonomic modules on a manifold $X$, parametrized by a curve $S$, the Hermitian duality functor (or conjugation functor) of Kashiwara. We prove that this functor is an equivalence with the similar category on the conjugate manifold $\overline X$, parametrized by the same curve. As a byproduct we introduce the notion of regular holonomic relative distribution.
In the setting of global geometric Langlands, we show that miraculous duality on the stack of principal bundles on a curve intertwines the functor of Poincare series with the dual functor to Whittaker coefficients. We construct, for arbitrary parabolic subgroups, Jacquet functors controlling constant terms of Poincare series and Whittaker coefficients of Eisenstein series in local terms.
Let X_1 and X_2 be schemes of finite type over a field of characteristic 0. Let Q be an object in the category D-mod(X_1\times X_2) and consider the functor F:D-mod(X_1)->Dmod(X_2) defined by Q. Assume that F admits a right adjoint also defined by an object P in D-mod(X_1\times X_2). The question that we pose and answer in this paper is how P is related to the Verdier dual of Q. We subsequently generalize this question to the case when X_1 and X_2 are no longer schemes but Artin stacks, where the situation becomes much more interesting.
The classification of affine line bundles on a compact complex space $X$ is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. For a fixed Chern class $c$, we introduce the affine Picard functor $Picaff_{X,x_0}^c:An^{op}\to Set$ which assigns to a complex space $T$ the set of families of linearly $x_0$-framed affine line bundles on $X$ with Chern class $c$ parameterized by $T$. Our main result states that this functor is representable if and only if the map $h^0:Pic^c(X)\to\mathbb{N}$ is constant. If this is the case, the …
To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitl…
In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham functor, sending logarithmic D-modules to certain graded sheaves on the so-called Kato-Nakayama space. For holonomic modules we show that the associated sheaves have finitely generated stalks and that the de Rham functor intertwines duality for D-modules with a version of Poincaré-Verdier duality on the Kato-Nakayama space. Finally, we explain how…
Let ${\mathfrak g}$ be a simple Lie algebra. For a level $κ$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_κ$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_κ(Gr_G)-mod$ be the category of $κ$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $Γ:D_κ(Gr_G)-mod\to {\mathfrak g}_κ-mod$. It is known that this functor is exact and faithful when $κ$ is negative or irrational. In this paper, we show that the functor $Γ$ is exact and…
This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of Beilinson's theorem that the vanishing cycle functor commutes with duality up to twist. Over an excellent base scheme, we show that the sliced nearby cycle functor commutes with duality up to modification of the base. We deduce that duality preserves universal local acyclicity over an excellent regular base. We also present Gabber's theorem that local acycl…
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we sho…
We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism and let $Z' \subseteq X$ and $Z \subseteq Y$ be closed subsets such that $f$ restricts to a proper morphism $f'\colon Z'\to Z$ of $f$. Then the functor $Rf_*$ is left adjoint to $RΓ_{Z'}f^!$ when considered as functors between complexes supported on $Z'$ or $Z$.
Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.
We show the compatibility between the moderate or nearby cycle functor for regular holonomic $\mathcal{D}$-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara.
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…
Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in "duality theories". In this paper we study and determine reflexive functors and we give many properties of reflexive functors.
Michigan State University（密歇根州立大学）；University of Kentucky（肯塔基大学）；University of Virginia（弗吉尼亚大学）；Indiana University（印第安纳大学）；Binghamton University（宾汉姆顿大学）
Let E be a locally free, rank n bimodule over a smooth projective scheme X, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of the internal Hom functor. When X is a smooth projective variety, we use the Gorensteinness of A to prove a version of Serre duality on Proj A, the non-commutative P^1 bundle defined by A.
The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings), satisfying the so called $α$-condition. For those pairings the induction functor is studied and a nice description of it is obtained. Along the paper several interestin…
The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine plane. Both schemes admit Białynicki-Birula decompositions into moduli spaces of ideals with prescribed lexicographic Gröbner deformations. We show that both decompositions are stratifications in the sense that the closure of each stratum is a union of certain other strata. We show that the corresponding two partial orderings on the set of of monomial ideals …
We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.
We construct a canonical pseudofunctor ^# on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor ^! to the full subcategory of the derived category having Cohen-Macaulay complexes as objects (a subcategory equivalent to the category of Cousin complexes, once a codimension function has been fixed). Specifically, for Cousin complexes M and any scheme map f:X -> Y as abov…
We attempt to determine the partition function of ${\cal N}=4$ super Yang-Mills theory for $ADE$ gauge groups on $K3$ and investigate the relation with affine Lie algebras. In particular we describe eta functions, which compose SU(N) partition function, by level $N$ $A_{N-1}$ theta functions. Moreover we find $D,E$ theta functions, which satisfy the Montonen-Olive duality for $D,E$ partition functions.
We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories.
Given a quasi-compact category fibered in groupoids $\mathcal{X}$ and a monoidal subcategory $\mathcal{C}$ of its category of locally free sheaves $\text{Vect}(\mathcal{X})$, we are going to introduce the stack of fiber functors $\text{Fib}_{\mathcal{X},\mathcal{C}}$ with source $\mathcal{C}$, which comes equipped with a map $\mathcal{P}_{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}_{\mathcal{X},\mathcal{C}}$ and a functor $\mathcal{G}\colon\mathcal{C}\to\text{Vect}(\text{Fib}_{\mathcal{X},\mathcal{C}})$. If $\mathcal{C}$ generates $\text{QCoh}(\mathcal{X})$ and $\mathcal{X}$ is an fpqc stack wi…
We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.
The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Bruguières. In one direction we have the usual contravariant functor which assigns to each Hopf algebroid its characters groupoid (the fiber groupoid at the ground field). In the other direction we construct the contravariant functor which associated to each discrete groupoid its Hopf algebroid of representative functions. This duality extends the well known duality between discrete groups and commu…
We show that the failure of the usual Verdier duality on Bun(G) leads to a new duality functor on the category of D-modules, and we study its relation to the operation of Eisenstein series.
We identify the category Shv_{Nilp}(Bun_G) of automorphic sheaves with nilpotent singular support with its own dual, and relate this structure to the Serre functor on Shv_{Nilp}(Bun_G) and miraculous duality.
Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra integrates to algebraic action of the Iwahori subgroup I. We study the action on this category of the convolution functors with the "central" sheaves on the affine flag scheme G((t))/I. We show that each object of our category is an "eigen-module" with respect to these functors. In order to prove this, we use the fusion product of modules over the affine Kac-Moody algebra.
In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…
We introduce a functor $\mathfrak{M}:\mathbf{Alg}\times\mathbf{Alg}^\mathrm{op}\rightarrow\mathrm{pro}\text{-}\mathbf{Alg}$ constructed from representations of $\mathrm{Hom}_\mathbf{Alg}(A,B\otimes ? )$. As applications, the following items are introduced and studied: (i) Analogue of the functor $π_0$ for algebras and affine schemes. (ii) Cotype of Weibel's concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $\mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homo…
We show that (quantum) D-modules of Givental type on affine flag varieties are transformed to D-modules of Beilinson-Drinfeld type on $n$-reduced Sato Grassmannians via affine Miura transformation.
We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal{A}$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal{A}$-modules and of $C$-comodules. When $\mathcal{A}$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(\!(t)\!)$-equivariant and $U^-(\!(t)\!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and a…
We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space of G-bundles on a complex curve X and flat G^L-bundles on X. Beilinson and Drinfeld construct it by applying a localization functor to representations of affine algebras of critical level. We show that in genus zero the corresponding D-modules are closely related to the diagonalization problem in the Gaudin model associated to G. This allows us to give a ne…
We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking t…
