arXiv subjects
Publications and source records attributed to Amnon Yekutieli.
This note contains a correction of the proofs of the main results of the paper [A. Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383-432]. The results are correct as originally stated.
In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper [YZ5]. In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper [Ye4] these results will be used to construct and study rigid dualizing complexes on schemes.
Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq_{B/A} M, which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ: M \iso Sq_{B/A} M then the pair (M,ρ) is called a rigid complex over B relative to A (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes. We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks f^{\flat}(M,ρ) (resp. f^{\sharp}(M,ρ)) along a finite (resp. essentially smooth) ring homomorphism f^* : B \to C. In a subsequent paper we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in a third paper, where we give a comprehensive version of Grothendieck duality for schemes. The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh.
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.
We present an averaging process for sections of a torsor under a unipotent group. This process allows one to integrate local sections of such a torsor into a global simplicial section. The results of this paper have applications to deformation quantization of algebraic varieties.
We introduce the notions of mixed resolutions and simplicial sections, and prove a theorem relating them. This result is used (in another paper) to study deformation quantization in algebraic geometry.
The purpose of this paper is to develop a suitable notion of continuous L_infinity morphism between DG Lie algebras, and to study twists of such morphisms.
Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by Htr D, is defined to be the injective dimension of the enveloping algebra of D. We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The main tool for the computation is Van den Bergh's rigid dualizing complex.
We study deformation quantizations of the structure sheaf O_X of a smooth algebraic variety X in characteristic 0. Our main result is that when X is D-affine, any formal Poisson structure on X determines a deformation quantization of O_X (canonically, up to gauge equivalence). This is an algebro-geometric analogue of Kontsevich's celebrated result.
A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated k-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when char k = 0. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.
A behavior is a closed shift invariant subspace of the space of sequences with entries in a field k. We work out an explicit duality for k-modules. This duality is then used to derive properties of behaviors, and their high dimensional and noncommutative generalizations.
Let A be a noncommutative noetherian with dualizing complex R. We study the multiplicities of indecomposable injectives in a minimal injective resolution of R. In particular when A is a Gorenstein ring we get information on minimal injective resolutions of A.
Let X be a scheme of finite type over a perfect field k. In this paper we study the relation between two important objects associated to X: the Grothendieck residue complex and the Beilinson adeles complex. It is known that the complex of adeles is a DGA (differential graded algebra). Our first main result is that the residue complex is a right DG module over the adeles complex. The second main result is that the de Rham residue complex is a DG module over the de Rham adeles complex. This action gives rise to the cap product in de Rham (co)homology.
Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects are even more complicated than in the commutative case, since they are complexes of bimodules. We develop methods to prove uniqueness, existence and functoriality of residue complexes. For a noetherian affine PI algebra over a field (admitting a noetherian connected filtration) we prove existence of the residue complex and describe its structure in detail.
Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C(X) of topological O_X-modules on X, called the complete Hochschild chain complex of X. To any O_X-module M - not necessarily quasi-coherent - we assign the complex Hom^{cont}_X(C(X),M) of continuous Hochschild cochains with values in M. Our first main result is that when X is smooth over K there is a functorial isomorphism between the complex of continuous Hochschild cochains and RHom_{X2}(O_X,M), in the derived category D(Mod(O_{X2})). The second main result is that if X is smooth of relative dimension n and n! is invertible in K, then the standard map from Hochschild chains to differential forms induces a decomposition of Hom^{cont}_X(C(X),M) in derived category D(Mod(O_X)). When M = O_X this is the precisely the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology with values in M.
The paper has been withdrawn by the author, due a gap in the proof of Theorem 6.1. The gap was discovered by M. Van den Bergh. Theorem 6.1 is used to prove the main result of the paper, namely Theorem 0.7 (decomposition in arbitrary characteristic). At this time we do not have an alternate proof. Other results of the paper (Theorems 0.2, 0.3, 0.6 and 3.1) are not effected by this problem, and they remain valid.
We prove that two-sided tilting complexes, and dualizing complexes, over simple Goldie rings (with some technical conditions) are always shifts of invertible bimodules. This allows us to describe the derived Picard groups of such rings, and to deduce these are Gorenstein (and sometimes even Auslander-Gorenstein Cohen-Macaulay) rings.
Let A be a finite dimensional algebra over an algebraically closed field K. The derived Picard group DPic(A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic(A) is a locally algebraic group, and its identity component is Out^0(A). If B is a derived Morita equivalent algebra then DPic(A) is isomorphic to DPic(B) as locally algebraic groups. Our results extend, and our based on, work of Huisgen-Zimmermann, Saorin and Rouquier.