Papers by Hvedri Inassaridze
Algebraic K-functors for $\Gamma$-rings
arXiv (Cornell University), Mar 1, 2023
European Journal of Mathematics
This is a further investigation of our approach to group actions in homological algebra in the se... more This is a further investigation of our approach to group actions in homological algebra in the settings of homology of Γsimplicial 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 ⋊ 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 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. Relations with extensions of crossed Γ-modules, in particular with relative extensions of group epimorphisms in the sense of Loday and with Γ-equivariant extensions of crossed Γ-modules are established. Universal and Γequivariant universal central Γ-extensions of Γ-perfect crossed Γmodules are constructed and Hopf formulas for the integral homology and Γ-equivariant integral homology of crossed Γ-modules are obtained. Finally, applications to algebraic K-theory, Galois theory of commutative rings and cohomological dimension of groups are given.
European Journal of Mathematics
This is a further investigation of our approach to group actions in homological algebra in the se... more 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 whe...
Given a thick subcategory of a triangulated category, we define a colocalisation and a natural lo... more 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. 2000 Mathematics Subject Classification. 18E30.
Non-Abelian Cohomology of Groups
gmj, 1997
Following Guin's approach to non-abelian cohomology [Guin, Pure Appl. Algebra 50: 109–137, 19... more Following Guin's approach to non-abelian cohomology [Guin, Pure Appl. Algebra 50: 109–137, 1988] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.
gmj, 1997
When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad.... more When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad. Sci. Paris 301: 337–340, 1985], [Guin, J. Pure Appl. Algebra 50: 109–137, 1988] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.
gmj, 1997
When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad.... more When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad. Sci. Paris 301: 337–340, 1985], [Guin, J. Pure Appl. Algebra 50: 109–137, 1988] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.
Journal of Homotopy and Related Structures, 2016
The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quas... more The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi-stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Fréchet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Fréchet algebras having a properly uniformly bounded approximate unit is established.
Topology and its Applications, 2005
We provide and study an equivariant theory of group (co)homology of a group G with coefficients i... more We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.
Mathematics, 2015
The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quas... more The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established. 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 and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Frechet algebras having a properly uniformly bounded approximate unit is established.
K-theory of special normed rings
Lecture Notes in Mathematics, 1990
Universal sequences of functors
Non-Abelian Homological Algebra and Its Applications, 1997
Our aim is to extend the well-known notion of satellites of additive functors to the non-additive... more Our aim is to extend the well-known notion of satellites of additive functors to the non-additive case using the universal proprety of satellites.
Higher K-functors
Algebraic K-Theory, 1995
Before defining the K-theory of exact categories we will need some results on the classifying spa... more Before defining the K-theory of exact categories we will need some results on the classifying space of a small category C.
Inassaridze. H. (Hvedri). 1932-Algebraic K-theory / by Hvedri Inassacidze. p. cm.-(Mathematics an... more Inassaridze. H. (Hvedri). 1932-Algebraic K-theory / by Hvedri Inassacidze. p. cm.-(Mathematics and Its applications v.311> Inc 1 udes bib 1 i ograph i ca 1 references and index.
Non-abelian homology and cohomology of groups
Non-Abelian Homological Algebra and Its Applications, 1997
In the first section some functorial propeties of the non-abelian tensor product of groups are es... more In the first section some functorial propeties of the non-abelian tensor product of groups are established. With the use of the non-abelian left derived functors [44,62] the homology groups of groups are constructed with coefficients in any group, as the left derived functors of the non-abelian tensor product, which generalize the classical theory of homology of groups. Exact sequences of the non-abelian homology groups and their application to algebraic K-theory of noncommutative local rings are given. Some sufficient conditions for the finiteness of the non-abelian tensor product of groups with non compatible actions are established, generalizing Ellis result [25]. These results are obtained by N.Inassaridze [51,52].
Tbilisi Mathematical Journal, 2014
Based on the concept of accessible subhemirings and inspired by the work on the general Kurosh-Am... more Based on the concept of accessible subhemirings and inspired by the work on the general Kurosh-Amitsur radical theory for rings, this paper studies the lower radical classes and the hereditary radical classes of hemirings. We characterize radical classes of hemirings, and con- struct a lower radical class from a homomorphically closed class. We provide a necessary and sufficient condition under which an upper radical class of hemirings becomes hereditary and prove that an upper radical class of a regular class of semirings is hereditary. Besides, we show that the Brown-McCoy radical class and a Jacobson-type radical class are hereditary.
Georgian Mathematical Journal - GEORGIAN MATH J, 1997
Following Guin's approach to non-abelian cohomology [4] and, using the notion of a crossed b... more Following Guin's approach to non-abelian cohomology [4] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2
Homology, Homotopy and Applications, 2005
It is proved that the homology and cohomology theories of groups and associative algebras are non... more It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative (n + 1)-th cohomology of groups is described in terms of n-fold extensions.
Glasgow Mathematical Journal, 2002
The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and f... more The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.
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Papers by Hvedri Inassaridze