Finitary Čech-de Rham Cohomology

The present paper continues (Mallios & Raptis, International Journal of Theoretical Physics, 2001, 40, 1885) and studies the curved finitary spacetime sheaves of incidence algebras presented therein from a ech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author's axiomatic approach to differential geometry via the theory of vector and algebra sheaves (Mallios, Geometry of Vector Sheaves: An Axiomtic Approach to Differential Geometry, Vols. 1–2, Kluwer, Dordrecht, 1998a; Mathematica Japonica (International Plaza), 1998b, 48, 93). The upshot of this study is that important classical differential geometric constructions and results usually thought of as being intimately associated with C-smooth manifolds carry through, virtually unaltered, to the finitary-algebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheaf-cohomology as developed in Mallios (1998a,b). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets (Raptis, International Journal of Theoretical Physics, 2000, 39, 1233; Mallios & Raptis, 2001), we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.     

Torsion Zero-Cycles on a Product of Canonical Lifts of Elliptic Curves

Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T(X) and T(Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T(X)/T(X)_div is conjectured to be finite, where T(X)_div is the maximal divisible subgroup of T(X). Furthermore, F(X) is conjectured to be isomorphic to T(Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.     

Bivariant cyclic theory

In this paper we construct a bivariant version of cyclic cohomology and study its fundamental properties. We prove universal coefficient theorems relating the bivariant theory with cyclic homology and cohomology, we construct products in the bivariant theory, and we analyse the notion of an HC-equivalence.Dedicated to Alexander Grothendieck     

Explicit weight two motivic cohomology complexes and algebraicK-theory

Beilinsonet al.'s motivic cohomology complex is related to a motivic complex (considered earlier by Lichtenbaum) formed using groups related to the scissors congruence groups. This involves an algebraic study of configurations of lines in projective spaces. One thereby obtains a more precise relationship between the cohomology of Beilinson's complex and lower dimensionalK-groups. Most of the proofs omitted in Beilinsonet al. are supplied here. There is also a list of certain torsional elements of the thirdK-group for fields between the rationals and the real cyclotomics.     

On the Picard group of a compact flat projective variety

In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap's work and Johnson's characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus covering the manifold. We prove a generalized version of the Appell-Humbert theorem which shows that the nontrivial elements of the Picard group are precisely those coming from the above construction. Our calculations finally give an estimate for the set of positive line bundles for such varieties.

     

Thomason's theorem for varieties over algebraically closed fields

We present a novel proof of Thomason's theorem relating Bott inverted algebraic -theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic -homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic -homology and algebraic -theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

K-theory tools for local and asymptotic cyclic cohomology

A generalization of the Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable -algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology, and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of -algebras. As another application, it is shown that these cyclic theories are rigid under Rieffel's deformation quantizations.

     

The first Cohomology of Affine $ \mathbb{Z}^{p} $ \mathbb{Z}^{p} -actions on Tori and applications to rigidity*

Let a minimal affine -action on the torus T ^q , p 2 and q 1. The cohomology of (see definition below) depends on both the algebraic properties of the induced action on H ¹(T ^q , ) and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to . Thus the -action F obtained by suspension of is parameter rigid, i.e., any other -action with the same orbit foliation is smoothly conjugate to a reparametrization of F by an automorphism of .*Partially supported by CNPq fellowship by Fondecyt Grant 1000047 and DGICT-UCN and fundación Andes, Chile.