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Amnon Yekutieli

Publications and source records attributed to Amnon Yekutieli.

At least 37 records · Page 2Linked to original sources
arXiv2015

Nonabelian Multiplicative Integration on Surfaces

We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-form, satisfying a certain differential equation. The geometric cycle of the integration is a kite in the pointed manifold. A kite is made up of a 2-dimensional simplex in the manifold, together with a path connecting this simplex to the base point of the manifold. The multiplicative integral is an element of the second Lie group in the crossed module. We prove several properties of the multiplicative integral. Among them is the 2-dimensional nonabelian Stokes Theorem, which is a generalization of Schlesinger's Theorem. Our main result is the 3-dimensional nonabelian Stokes Theorem. This is a totally new result. The methods we use are: the CBH Theorem for the nonabelian exponential map; piecewise smooth geometry of polyhedra; and some basic algebraic topology. The motivation for this work comes from twisted deformation quantization and descent for nonabelian gerbes. Similar questions arise in nonabelian gauge theory.

math.DG↗
arXiv2015

Introduction to Derived Categories

Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors between abelian categories) was established. This "new" homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the "old" homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the "old" homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript "A Course on Derived Categories" (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.

math.KT↗
arXiv2014

Local Beilinson-Tate Operators

In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions. Our paper presents a variant of Beilinson's operator-theoretic construction. We consider an n-dimensional topological local field K, and define a ring of operators E(K) that acts on K, which we call the ring of local Beilinson-Tate operators. Our definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). We study various properties of the ring E(K). In particular we show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. Presumably this residue functional coincides with the residue functional that we had constructed in 1992; but we leave this as a conjecture.

math.NT↗
arXiv2014

Twisted Deformation Quantization of Algebraic Varieties

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.

math.AG↗
arXiv2014

Another Proof of a Theorem of Van den Bergh about Graded-Injective Modules

M. Van den Bergh proved that if A is a left noetherian N-graded ring, and I is a graded-injective left A-module, then the injective dimension of I in the ungraded sense is at most 1. In this note we give another proof of this result. Our proof follows the same strategy as the original proof by Van den Bergh, yet it is somewhat more conceptual, in that it isolates the precise "reason" for the dimension jump.

math.RA↗
arXiv2014

Combinatorial Descent Data for Gerbes

We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data.

math.KT↗
arXiv2013

Cohomologically Cofinite Complexes

Let A be a commutative noetherian ring, and \a an ideal in it. In this paper we continue the study, begun in [PSY1], of the derived \a-adic completion and the derived \a-torsion functors. Here are our results: (1) a structural characterization of bounded above cohomologically complete complexes; (2) the Cohomologically Complete Nakayama Theorem; and (3) a characterization of cohomologically cofinite complexes.

math.AC↗
arXiv2013

Completion by Derived Double Centralizer

Let A be a commutative ring, and let \a be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically \a-torsion complexes. We prove that the derived double centralizer of M is isomorphic to the \a-adic completion of A. The proof relies on the MGM equivalence from [PSY] and on derived Morita equivalence. Our result extends earlier work of Dwyer-Greenlees-Iyengar [DGI] and Efimov [Ef].

math.AC↗
arXiv2012

On the Homology of Completion and Torsion

Let A be a commutative ring, and \a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically \a-adically complete complexes and the category of cohomologically \a-torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso- Jeremias-Lipman, Schenzel and Dwyer-Greenlees.

math.AC↗
arXiv2012

Deformations of Affine Varieties and the Deligne Crossed Groupoid

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex. The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.

math.AG↗
arXiv2012

MC Elements in Pronilpotent DG Lie Algebras

Consider a pronilpotent DG (differential graded) Lie algebra over a field of characteristic 0. In the first part of the paper we introduce the reduced Deligne groupoid associated to this DG Lie algebra. We prove that a DG Lie quasi-isomorphism between two such algebras induces an equivalence between the corresponding reduced Deligne groupoids. This extends the famous result of Goldman- Millson (attributed to Deligne) to the unbounded pronilpotent case. In the second part of the paper we consider the Deligne 2-groupoid. We show it exists under more relaxed assumptions than known before (the DG Lie algebra is either nilpotent or of quasi quantum type). We prove that a DG Lie quasi-isomorphism between such DG Lie algebras induces a weak equivalence between the corresponding Deligne 2-groupoids. In the third part of the paper we prove that an L-infinity quasi-isomorphism between pronilpotent DG Lie algebras induces a bijection between the sets of gauge equivalence classes of Maurer-Cartan elements. This extends a result of Kontsevich and others to the pronilpotent case.

math.QA↗
arXiv2011

Twisted Deformation Quantization of Algebraic Varieties (Survey)

Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes.

math.AG↗
arXiv2010

Central Extensions of Gerbes

We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in a subsequent paper to study twisted deformation quantization on algebraic varieties.

math.AG↗
arXiv2008

Rigid Dualizing Complexes via Differential Graded Algebras (Survey)

In this article we survey recent results on rigid dualizing complexes over commutative algebras. We begin by recalling what are dualizing complexes. Next we define rigid complexes, and explain their functorial properties. Due to the possible presence of torsion, we must use differential graded algebras in the constructions. We then discuss rigid dualizing complexes. Finally we show how rigid complexes can be used to understand Cohen-Macaulay homomorphisms and relative dualizing sheaves.

math.AG↗