arXiv subjects

Amnon Yekutieli

Publications and source records attributed to Amnon Yekutieli.

63 records · Page 4Linked to original sources

arXiv2000

Derived Picard Groups of Finite Dimensional Hereditary Algebras

Let A be an algebra over a field k, and denote by D^b(Mod A) the bounded derived category of left A-modules. The derived Picard group DPic_k(A) is the group of triangle auto-equivalences of D^b(Mod A) induced by tilting complexes. We study the group DPic_k(A) when A = k Δis the path algebra of a finite quiver Δ. We obtain general results on the structure of DPic_k(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic_k(A) on a certain infinite quiver. This representation is faithful whenΔis a tree, and then DPic_k(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic_k(A). When A is hereditary, DPic_k(A) coincides with the full group of k-linear triangle auto-equivalences of D^b(Mod} A). Hence we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b(Mod A). These include the derived categories of certain noncommutative spaces introduced by Kontsevich-Rosenberg.

math.RA↗

arXiv1998

Adelic Chern Forms and Applications

Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms c_{i}(E;\nabla) in Γ(X, \cal{A}^{2i}_{X}). Here \cal{A}^{.}_{X} is the sheaf Beilinson adeles and \nabla is an adelic connection. When X is smooth these adeles calculate the algebraic De Rham cohomology, and c_{i}(E) = [c_{i}(E;\nabla)] are the usual Chern classes. We include three applications of the construction: (1) existence of adelic secondary (Chern-Simons) characteristic classes on any smooth X and any vector bundle E; (2) proof of the Bott Residue Formula for a vector field action; and (3) proof of a Gauss-Bonnet Formula on the level of differential forms, namely in the De Rham-residue complex.

math.AG↗

arXiv1998

Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the derived tensor product with T is an equivalence. The complex T is called a tilting complex. When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. So we can compute DPic(A) in these cases. Assume A is noetherian. Dualizing complexes over A were defined by the author some years ago. These are complexes of bimodules which generalize the commutative definition of Grothendieck. We prove that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. Finally we consider finite k-algebras. For the algebra A of upper triangular 2 x 2 matrices over k, we prove that t^{3} = s, where t, s are the classes in DPic(A) of Hom_{k}(A, k) and A[1] respectively. In the Appendix by Elena Kreines this result is generalized to upper triangular n x n matrices, and it is shown that the relation t^{n + 1} = s^{n - 1} holds.

math.RA↗

arXiv1998

The Rigid Dualizing Complex of a Universal Enveloping Algebra

Let k be a field and A a noetherian (noncommutative) k-algebra. The rigid dualizing complex of A was introduced by Van den Bergh. When A = U(g), the enveloping algebra of a finite dimensional Lie algebra g, Van den Bergh conjectured that the rigid dualizing complex is (U(g) \otimes \wedge^{n} g)[n], where n = dim g. We prove this conjecture, and give a few applications in representation theory and Hochschild cohomology.

math.RA↗

arXiv1998

Rings with Auslander Dualizing Complexes

A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes.

math.RA↗

arXiv1998

Smooth Formal Embeddings and the Residue Complex

Let π: X -> S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X -> \frak{X}, where \frak{X} is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion \frak{X} = Y_{/X} where X \subset Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham (co)homology. Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of π^{!} \cal{O}_{S}. We start with Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, δ) and the residue complex of Grothendieck. When πis equidimensional of dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi. Another issue we discuss is Grothendieck Duality on a noetherian formal scheme \frak{X}. Our results on duality are used in the construction of \cal{K}^{.}_{X/S}.

alg-geom↗

arXiv1998

Residues and Differential Operators on Schemes

Beilinson Completion Algebras (BCAs) are generalizations of complete local rings, and have a rich algebraic-analytic structure. These algebras were introduced in my paper "Traces and Differential Operators over Beilinson Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are used to give an explicit construction of the Grothendieck residue complex on an algebraic scheme. This construction reveals new properties of the residue complex, and in particular its interaction with differential operators. Applications include: (i) results on the algebraic structure of rings of differential operators; (ii) an analysis of the niveau spectral sequence of De Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t. etale morphisms; (iv) an algebraic description of the intersection cohomology D-module of a curve.

alg-geom↗

arXiv1995

Some Remarks on Beilinson Adeles

This paper contains two remarks on Beilinson's adeles with values in the De Rham complex of a scheme. The first is an interpretation, in terms of adeles, of the decomposition of the De Rham complex on a scheme defined modulo $p^{2}$ (the result of Deligne-Illusie). The second remark is about the possible relation between adeles and Hodge decomposition. We work out a counter example.

alg-geom↗

arXiv1995

Traces and Differential Operators over Beilinson Completion Algebras

A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a perfect field k, whose residue fields are high dimensional local fields. In addition A is a semi-topological algebra. The completion of the structure sheaf of an algebraic k-variety along a saturated chain of points is the prototypical example of a BCA. We single out two kinds of homomorphisms between BCAs: morphisms and intensifications. The first kind includes residually finite local homomorphisms, whereas the second kind is a sort of localization. We prove that every BCA A has a dual module K(A), and these dual modules are contravariant w.r.t. morphisms and covariant w.r.t. intensifications. For any semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M, K(A)). This duality operation has the remarkable property of respecting differential operators: given a continuous DO D : M --> N, there is a dual DO Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on any finite type k-scheme X, and to derive many of its properties.

alg-geom↗