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April 2006


Non-commutative Henselian Rings.

Masood Aryapoor

Non-commutative Henselian rings are defined and it is shown that a local ringwhich is complete and separated in the topology defined by its maximal ideal isHenselian provided that it is almost commutative.

Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes.

Amnon Yekutieli, James J. Zhang

A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A isa noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes overquasi-coherent ringed schemes and study their properties. For a separateddifferential quasi-coherent ringed scheme of finite type over a field, we proveexistence and uniqueness of a rigid dualizing complex. In the proof we use thetheory of perverse coherent sheaves in order to glue local pieces of the rigiddualizing complex into a global complex.

A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra.

Nikolai Durov, Stjepan Meljanac, Andjelo Samsarov, Zoran Skoda

Given a $n$-dimensional Lie algebra $g$ over a field $k \supset \mathbb Q$,together with its vector space basis $X^0_1,…, X^0_n$, we give a formula,depending only on the structure constants, representing the infinitesimalgenerators, $X_i = X^0_i t$ in $g\otimes_k k [[t]]$, where $t$ is a formalvariable, as a formal power series in $t$ with coefficients in the Weyl algebra$A_n$. Actually, the theorem is proved for Lie algebras over arbitrary rings$k\supset Q$.

We provide three different proofs, each of which is expected to be useful forgeneralizations. The first proof is obtained by direct calculations withtensors. This involves a number of interesting combinatorial formulas instructure constants. The final step in calculation is a new formula involvingBernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions ofcertain spaces of tensors are also calculated. The second method of proof isgeometric and reduces to a calculation of formal right-invariant vector fieldsin specific coordinates, in a (new) variant of formal group scheme theory. Thethird proof uses coderivations and Hopf algebras.

Introducing Crystalline Graded Algebras.

Erna Nauwelaerts, Freddy Van Oystaeyen

We introduce a new class of graded rings extending the class of generalizedWeyl algebras. These rings are orders in crossed products of the most generaltype, and we introduce their basic structure theory. We provide an extensivelist of examples, some completely new but also some considered earlier, andhighlight some specific structure results. Many new and interesting problemsabout the crystalline graded rings may be identified.

Rigid Complexes via DG Algebras.
Amnon Yekutieli, James J. Zhang

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 :MisoSq B /AM 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 (M,ρ)(resp.f (M,ρ)) along a finite (resp. essentially smooth) ring homomorphism f * :B→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.

D-Bundles and Integrable Hierarchies.

David Ben-Zvi, Thomas Nevins

We study the geometry of D-bundles–locally projective D-modules–onalgebraic curves, and apply them to the study of integrable hierarchies,specifically the multicomponent Kadomtsev-Petviashvili (KP) and spinCalogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometricdescription as flows on moduli spaces of D-bundles; in particular, we provethat the local structure of D-bundles is captured by the full SatoGrassmannian. The rational, trigonometric, and elliptic solutions of KP aretherefore captured by D-bundles on cubic curves E, that is, irreducible(smooth, nodal, or cuspidal) curves of arithmetic genus 1. We develop aFourier-Mukai transform describing D-modules on cubic curves E in terms of(complexes of) sheaves on a twisted cotangent bundle over E. We then apply thistransform to classify D-bundles on cubic curves, identifying their modulispaces with phase spaces of general CM particle systems (realized through thegeometry of spectral curves in our twisted cotangent bundle). Moreover, it isimmediate from the geometric construction that the flows of the KP and CMhierarchies are thereby identified and that the poles of the KP solutions areidentified with the positions of the CM particles. This provides a geometricexplanation of a much-explored, puzzling phenomenon of the theory of integrablesystems: the poles of meromorphic solutions to KP soliton equations moveaccording to CM particle systems.

Noncommutative Images of Commutative Spectra.

Edward S. Letzter

We initiate a unified, axiomatic study of noncommutative algebras R whoseprime spectra are, in a natural way, finite unions of commutative noetherianspectra. Our results illustrate how these commutative spectra can befunctorially “sewn together” to form Spec R. In particular, we construct abimodule-determined functor Mod Z -> Mod R, for a suitable commutativenoetherian ring Z, from which there follows a finite-to-one, continoussurjection Spec Z -> Spec R. Algebras satisfying the given axiomatic frameworkinclude PI algebras finitely generated over fields, noetherian PI algebras,enveloping algebras of complex finite dimensional solvable Lie algebras,standard generic quantum semisimple Lie groups, quantum affine spaces,quantized Weyl algebras, and standard generic quantizations of the coordinatering of nxn matrices. In all of these examples (except for thenon-finitely-generated noetherian PI algebras), Z is finitely generated over afield, and the constructed map of spectra restricts to a surjection Max Z ->Prim R.

Commutator Hopf subalgebras and irreducible representations.

Edward S. Letzter

S. Montgomery and S. Witherspoon proved that upper and lower semisolvable,semisimple, finite dimensional Hopf algebras are of Froebenius type when theirdimensions are not divisible by the characteristic of the base field. In thisnote we show that a finite dimensional, semisimple, lower solvable Hopf algebrais always of Froebenius type, in arbitrary characteristic.

Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras.

Kenneth A. Brown (Glasgow), James J. Zhang (Washington)

We show that many noetherian Hopf algebras A have a rigid dualising complex Rwith R isomorphic to ^{\nu}A^1 [d]. Here, d is the injective dimension of thealgebra and \nu is a certain k-algebra automorphism of A, unique up to an innerautomorphism. In honour of the finite dimensional theory which is herebygeneralised we call \nu the Nakayama automorphism of A. We prove that \nu =S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphismof A determined by the left integral of A. The Hochschild homology andcohomology groups with coefficients in a suitably twisted free bimodule areshown to be non-zero in the top dimension d, when A is an Artin-Schelterregular noetherian Hopf algebra of global dimension d. (Twisted) Poincareduality holds in this setting, as is deduced from a theorem of Van den Bergh.Calculating \nu for A using also the opposite coalgebra structure, we determinea formula for S^4 generalising a 1976 formula of Radford for A finitedimensional. Applications of the results to the cases where A is PI, anenveloping algebra, a quantum group, a quantised function algebra and a groupalgebra are outlined.

The relative Riemann-Roch theorem from Hochschild homology.

Ajay C. Ramadoss

This write up attempts to clarify a preprint by Markarian [2] which provesthe relative Riemann-Roch theorem starting from a theorem essentiallydescribing how the Hochschild Kostant Rosenberg (HKR) map from the completedHochschild chain complex of a smooth scheme X over a field of characteristic0 fails to respect co multiplication. We attempt to elaborate on the core stepsin [2]. In the process, we obtain a proof of the compatibility of the HKR maptwisted by the square root of the Todd genus with a version of the generalizedMukai pairing defined by Caldararu [3]. This settles a part of a conjecture ofCaldararu [3] regarding the equivalence between the Hochschild and Hodgestructures of a smooth complex scheme.

Double Poisson algebras.

Michel Van den Bergh

In this paper we develop Poisson geometry for non-commutative algebras. Thisgeneralizes the bi-symplectic geometry which was recently, and independently,introduced by Crawley-Boevey, Etingof and Ginzburg.

Our (quasi-)Poisson brackets induce classical (quasi-)Poisson brackets onrepresentation spaces. As an application we show that the moduli spaces ofrepresentations associated to the deformed multiplicative preprojectivealgebras recently introduced by Crawley-Boevey and Shaw carry a natural Poissonstructure.