179 research outputs found
On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes
We show that if is an infinitesimal elementary supergroup scheme of
height , then the cohomological spectrum of is naturally
homeomorphic to the variety of supergroup homomorphisms
from a certain (non-algebraic) affine
supergroup scheme into . In the case , we further
identify the cohomological support variety of a finite-dimensional
-supermodule as a subset of . We then discuss how our
methods, when combined with recently-announced results by Benson, Iyengar,
Krause, and Pevtsova, can be applied to extend the homeomorphism
to arbitrary infinitesimal unipotent supergroup
schemes.Comment: Fixed some algebra misidentifications, primarily in Sections 1.3 and
3.3. Simplified the proof of Proposition 3.3.
Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture
We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is
the first step of Voevodsky's proof of this conjecture for arbitrary prime l)
in a rather clear and elementary way. Assuming this conjecture, we construct a
6-term exact sequence of Galois cohomology with cyclotomic coefficients for any
finite extension of fields whose Galois group has an exact quadruple of
permutational representations over it. Examples include cyclic groups, dihedral
groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4.
Several exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn
are proven in this way. In addition, we introduce a more sophisticated version
of the classical argument known as "Bass-Tate lemma". Some results about
annihilator ideals in Milnor rings are deduced as corollaries.Comment: LaTeX 2e, 17 pages. V5: Updated to the published version + small
mistake corrected in Section 5. Submitted also to K-theory electronic
preprint archives at http://www.math.uiuc.edu/K-theory/0589
Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry
The purpose of this paper is to provide the reader with a collection of
results which can be found in the mathematical literature and to apply them to
hyperbolic spaces that may have a role in physical theories. Specifically we
apply K-theory methods for the calculation of brane charges and RR-fields on
hyperbolic spaces (and orbifolds thereof). It is known that by tensoring
K-groups with the rationals, K-theory can be mapped to rational cohomology by
means of the Chern character isomorphisms. The Chern character allows one to
relate the analytic Dirac index with a topological index, which can be
expressed in terms of cohomological characteristic classes. We obtain explicit
formulas for Chern character, spectral invariants, and the index of a twisted
Dirac operator associated with real hyperbolic spaces. Some notes for a
bivariant version of topological K-theory (KK-theory) with its connection to
the index of the twisted Dirac operator and twisted cohomology of hyperbolic
spaces are given. Finally we concentrate on lower K-groups useful for
description of torsion charges.Comment: 26 pages, no figures, LATEX. To appear in the Classical and Quantum
Gravit
Non-trivial stably free modules over crossed products
We consider the class of crossed products of noetherian domains with
universal enveloping algebras of Lie algebras. For algebras from this class we
give a sufficient condition for the existence of projective non-free modules.
This class includes Weyl algebras and universal envelopings of Lie algebras,
for which this question, known as noncommutative Serre's problem, was
extensively studied before. It turns out that the method of lifting of
non-trivial stably free modules from simple Ore extensions can be applied to
crossed products after an appropriate choice of filtration. The motivating
examples of crossed products are provided by the class of RIT algebras,
originating in non-equilibrium physics.Comment: 13 page
The quotient Unimodular Vector group is nilpotent
Jose-Rao introduced and studied the Special Unimodular Vector group
and , its Elementary Unimodular Vector subgroup. They
proved that for , is a normal subgroup of . The
Jose-Rao theorem says that the quotient Unimodular Vector group,
, for , is a subgroup of the orthogonal quotient
group . The latter group is known to be
nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is
the former.
In this article we give a direct proof, following ideas of A. Bak, to show
that the quotient Unimodular Vector group is nilpotent of class . We also use the Quillen-Suslin theory, inspired by A. Bak's method,
to prove that if , with a local ring, then the quotient
Unimodular Vector group is abelian
Framed transfers and motivic fundamental classes
We relate the recognition principle for infinite P1-loop spaces to the theory of motivic fundamental classes of Deglise, Jin and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with An/(An-0), and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite R-correspondences for R a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calmes and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of R-module spectra factors through the category of finite R-correspondences
Algebraic K-theory of endomorphism rings
We establish formulas for computation of the higher algebraic -groups of
the endomorphism rings of objects linked by a morphism in an additive category.
Let be an additive category, and let Y\ra X be a covariant
morphism of objects in . Then for all , where is the
quotient ring of the endomorphism ring of modulo the
ideal generated by all those endomorphisms of which factorize through .
Moreover, let be a ring with identity, and let be an idempotent element
in . If is homological and has a finite projective resolution
by finitely generated projective -modules, then for all . This reduces calculations of the higher
algebraic -groups of to those of the quotient ring and the corner
ring , and can be applied to a large variety of rings: Standardly
stratified rings, hereditary orders, affine cellular algebras and extended
affine Hecke algebras of type .Comment: 21 pages. Representation-theoretic methods are used to study the
algebraic K-theory of ring
Motivic Eilenberg-Maclane spaces
This paper is the second one in a series of papers about operations in
motivic cohomology. Here we show that in the context of smooth schemes over a
field of characteristic zero all the bi-stable operations can be obtained in
the usual way from the motivic reduced powers and the Bockstein homomorphism.Comment: This version is very close to the final version accepted to the
publication in Publ. IHE
Construction of a modified perfect form of a system of residual classes based on the vieta’s theorem
Dilogarithm Identities in Conformal Field Theory and Group Homology
Recently, Rogers' dilogarithm identities have attracted much attention in the
setting of conformal field theory as well as lattice model calculations. One of
the connecting threads is an identity of Richmond-Szekeres that appeared in the
computation of central charges in conformal field theory. We show that the
Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be
interpreted as a lift of a generator of the third integral homology of a finite
cyclic subgroup sitting inside the projective special linear group of all real matrices viewed as a {\it discrete} group. This connection
allows us to clarify a few of the assertions and conjectures stated in the work
of Nahm-Recknagel-Terhoven concerning the role of algebraic -theory and
Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related
to hyperbolic 3-manifolds as suggested but is more appropriately related to the
group manifold of the universal covering group of the projective special linear
group of all real matrices viewed as a topological group. This
also resolves the weaker version of the conjecture as formulated by Kirillov.
We end with the summary of a number of open conjectures on the mathematical
side.Comment: 20 pages, 2 figures not include
- …