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Teimuraz Pirashvili

Publications and source records attributed to Teimuraz Pirashvili.

At least 19 recordsLinked to original sources
arXiv2021

Symmetric cohomology of groups and Poincaré duality

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_λ(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_λ(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.

math.GR↗
arXiv2019

A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann

Our (weak) conjecture claims that a finite dimensional Lie algebra ${\bf g}$ over the field of complex numbers is semi-simple iff the Leibniz homology vanishes in positive dimensions $HL_i({\bf g})=0$, $i>0$. We will indicate a mistake in the recent proof of this conjecture due to Burde and Wagemann.

math.KT↗
arXiv2019

A subcomplex of Leibniz complex

Using the free graded Lie algebras we introduce a natural subcomlex of the Loday's complex of a Leibniz algebra. Our conjecture says, that for free Leibniz algebras, the complex is acyclic.

math.KT↗
arXiv2014

Cohomologie des foncteurs polynomiaux sur les groupes libres

We show that extension groups between two polynomial functors on free groups are the same in the category of all functors and in a subcategory of polynomial functors of bounded degree. We give some applications. ---- On montre que les groupes d'extensions entre foncteurs polynomiaux sur les groupes libres sont les mêmes dans la catégorie de tous les foncteurs et dans une sous-catégorie de foncteurs polynomiaux de degré borné. On donne quelques applications.

math.KT↗
arXiv2012

Polynomial functors from Algebras over a set-operad and non-linear Mackey functors

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations.

math.AT↗
arXiv2011

Cohomology with coefficients in stacks of Picard categories

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and the Ext groups of appropriate stacks.

math.AT↗
arXiv2010

On abelian 2-categories and derived 2-functors

This is an extended version of my earlier articel "Projective and injective objects in symmetric categorical groups. arXiv:1007.0121v1." Several new facts added, including the material on the derived 2-functors and the proof of the Gabriel-Mitchel theorem and the Morita theory for 2-rings.

math.CT↗
arXiv2010

Projective and injective objects in symmetric categorical groups

We prove that the 2-category of symmetric categorical groups have enough projective and injective objects. This was conjectured by Bourn and Vitale in 2002 and was anounced recently by Fang Huang, Shao-Han Chen, Wei Chen and Zhu-Jun Zheng in arXiv:1006.4677 with wrong proof.

math.CT↗
arXiv2009

Categorical rings subsume ann-categories

The article has been withdrawn - the proof of the final corollary turned out to rely on an unproven statement. If the authors will manage to repair the argument, resubmission will be made.

math.CT↗
arXiv2008

Abelian categories versus abelian 2-categories

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.

math.CT↗
arXiv2008

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander and strict polynomial functors in the sense of Friedlander and Suslin. Applications to functor cohomology are discussed.

math.RT↗
arXiv2006

Universal coefficient theorem in triangulated categories

Let T be a triangulated category, A a graded abelian category and h: T -> A a homology theory on T with values in A. If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category and the ideal Ker(h) is a square zero ideal which as a bifunctor on T is isomorphic to Ext^1_A(h(-)[1], h(-)).

math.CT↗
arXiv2006

Third Mac Lane cohomology via categorical rings

It is proved that the third Mac Lane cohomology group of a ring R with coefficients in a bimodule B classifies categorical rings having R as the ring of isomorphism classes of objects and B as the bimodule of automorphisms of the neutral object.

math.KT↗
arXiv2005

Quadratic envelope of the category of class two nilpotent groups

We exhibit an extension of the category of class two nilpotent groups. It has the same objects but, unlike the latter, its morphisms are closed under pointwise addition of maps. At the same time the class of its morphisms is much smaller than the class of all maps. In fact, the morphisms are quadratic maps of very special kind. It is indicated that classification of various class two nilpotent groups up to isomorphism in this larger category can give interesting results.

math.GR↗
arXiv2005

Cohomology of the Grothendieck construction

We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any funtor L: K -> CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction of L in terms of the cohomology of K and of L(k), for k an object in K.

math.CT↗