arXiv subjects
Publications and source records attributed to Hvedri Inassaridze.
This is an attempt to extend to algebraic K-theory our approach to group actions in homological algebra that could be called an introduction to $Γ$-algebraic K-theory. For $Γ$-rings the Milnor algebraic K-theory and Swan's algebraic K-functors are introduced and investigated. Particularly the Matsumoto conjecture related to the symbol group, and the Milnor conjectures related to Witt and Chow groups are extended and proven.
This is a further investigation of our approach to group actions in homological algebra in the settings of homology of Γ-simplicial groups, particularly of Γ-equivariant homology and cohomology of Γ-groups. This approach could be called Γ-homological algebra. The abstract kernel of non-abelian extensions of groups, its relation with the obstruction to the existence of non-abelian extensions and with the second group cohomology are extended to the case of non-abelian Γ-extensions of Γ-groups. We compute the rational Γ-equivariant (co)homology groups of finite cyclic Γ-groups. The isomorphism of the group of n-fold Γ-equivariant extensions of a Γ-group G by a G o Γ-module A with the (n+1)th Γ-equivariant group cohomology of G with coefficients in A is proven.We define the Γ-equivariant Hochschild homology as the homology of the Γ- Hochschild complex involving the cyclic homology when the basic ring contains rational numbers and generalizing the Γequivariant(co)homology of Γ-groups when the action of the group Γ on the Hochschild complex is induced by its action on the basic ring. Important properties of the Γ-equivariant Hochschild homology related to Kahler differentials, Morita equivalence and derived functors are established. Group (co)homology and Γ-equivariant group (co)homology of crossed Γ-modules are introduced and investigated by using relevant derived functors Finally, applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups are given.
The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established and we prove that the Quillen K- groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Frechet algebras having a properly uniformly bounded approximate unit. Based on these results the K-regularity property for quasi-stable Frechet algebras having a properly uniformly bounded approximate unit is established.
Given a set of prime numbers S, we localise equivariant bivariant Kasparov theory at S and compare this localisation with Kasparov theory by an exact sequence. More precisely, we define the localisation at S to be KK^G(A,B) tensored with the ring of S-integers Z[S^-1]. We study the properties of the resulting variants of Kasparov theory.
Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.
We develop a finite KKG-theory of C*-algebras following Arlettaz- H.Inassaridze's approach to finite algebraic K-theory. The Browder- Karoubi-Lambre's theorem on the orders of the elements for finite algebraic K-theory is extended to finite KKG-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost all KKG-theory properties and one has a long exact sequence relating KK-theory, rational bivariant K-theory and torsion KK-theory. For a given homology theory on the category of separable GC*-algebras finite, rational and torsion homology theories are introduced and investigated. In particular, we formulate finite, torsion and rational versions of Baum-Connes Conjecture. The later is equivalent to the investigation of rational and q-finite analogues for Baum-Connes Conjecture for all prime q.