On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms $\rho: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further identify the cohomological support variety of a finite-dimensional $G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. 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 $\mathcal{N}_r(G) \cong |G|$ 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 $SUm_r(R)$ and $EUm_r(R)$, its Elementary Unimodular Vector subgroup. They proved that for $r \geq 2$, $EUm_r(R)$ is a normal subgroup of $SUm_r(R)$. The Jose-Rao theorem says that the quotient Unimodular Vector group, $SUm_r(R)/EUm_r(R)$, for $r \geq 2$, is a subgroup of the orthogonal quotient group $SO_{2(r+1)}(R)/EO_{2(r + 1)}(R)$. 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 $\leq d = \dim(R)$. We also use the Quillen-Suslin theory, inspired by A. Bak's method, to prove that if $R = A[X]$, with $A$ 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 $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let Y\ra X be a covariant morphism of objects in ${\mathcal C}$. Then $K_n\big(_{\mathcal C}(X\oplus Y)\big)\simeq K_n\big(_{{\mathcal C},Y}(X)\big)\oplus K_n\big(_{\mathcal C}(Y)\big)$ for all $1\le n\in \mathbb{N}$, where $_{{\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factorize through $Y$. Moreover, let $R$ be a ring with identity, and let $e$ be an idempotent element in $R$. If $J:=ReR$ is homological and $_RJ$ has a finite projective resolution by finitely generated projective $R$-modules, then $K_n(R)\simeq K_n(R/J)\oplus K_n(eRe)$ for all $n\in \mathbb{N}$. This reduces calculations of the higher algebraic $K$-groups of $R$ to those of the quotient ring $R/J$ and the corner ring $eRe$, 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 $\tilde{A}$.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

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 $2 \times 2$ 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 $K$-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 $2 \times 2$ 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