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For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to \mathrm{Ab}$. Applications involve K\"unneth theorem for higher limits and ${\sf lim}$-finiteness of ${\bf fr}$-codes. A dictionary for the ${\bf fr}$-codes with words of length $\leq 3$ is given., Comment: 23 pages

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.

We give an overview of five rationalization theories for spaces (Bousfield-Kan's $\mathbb Q$-completion; Sullivan's rationalization; Bousfield's homology rationalization; Casacuberta-Peschke's $\Omega$-rationalization; G\'{o}mez-Tato-Halperin-Tanr\'{e}'s $\pi_1$-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups ($\mathbb Q$-completion; $H\mathbb Q$-localization; Baumslag rationalization) that extend the classical Malcev completion.

Для сильно связной категории $\mathscr C$ с попарными копроизведениями определен косимплициальный объект, служащий своего рода резольвентой для вычисления высших производных функторов функтора предела $\lim\colon\mathrm{Ab}^{\mathscr C} {\to} \mathrm{Ab}$. В качестве приложений получена формула Кюннета для высших пределов и $\lim$-конечность $\mathbf{fr}$-кодов. Также вычислен словарь для $\mathbf{fr}$-кодов со словами длины $\leq 3$. Библиография: 19 названий.

A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory it is shown that the third reduced $K$-functor can be defined as the colimit of the second reduced $K$-functor applied to the fibre square of a free presentation of an algebra. A Hopf-type formula for odd dimensional cyclic homology of an algebra over a field of characteristic zero is also proved., Comment: 1 figure

We prove that for a subring $R\subseteq \mathbb Q$ and a free group $F$ of rank at least $2$ the length of the Bousfield's $HR$-localization tower for $F$ is at least $\omega+\omega$. The key ingredient of the proof is the theory of polynomial functors over $\mathbb Q.$

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over Z , and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module M over a principal ideal domain that connects the exterior and the symmetric powers 0 → Λ n M → M ⊗ Λ n − 1 M → … → S n − 1 M ⊗ M → S n M → 0 is purely acyclic.

Развита теория высших копределов по категориям копредставлений. Показано, что различные гомологические функторы такие, как хохшильдовы и циклические гомологии алгебр над полем характеристики нуль, симплициальные производные функторы и гомологии групп, могут быть получены как высшие копределы некоторых простых функторов. Точная последовательность Конна, соединяющая хохшильдовы и циклические гомологии, получена с использованием этого подхода как следствие простой короткой точной последовательности. В качестве приложения развитой техники показано, что третий редуцированный $K$-функтор может быть определен как копредел второго редуцированного $K$-функтора, примененного к расслоенному квадрату копредставления алгебры. Также доказана формула типа Хопфа для четномерных циклических гомологий алгебры над полем характеристики нуль. Библиография: 17 названий.

In this paper, the Hℤ-length of different groups is studied. By definition, this is the length of the Hℤ-localization tower or the length of the transfinite lower central series of Hℤ-localization. It is proved that, for a free noncyclic group, its Hℤ-length is ≥ ω+2. For a large class of ℤ[C]-modules M, where C is an infinite cyclic group, it is proved that the Hℤ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its Hℤ-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M⋊C is finitely presented and H2(M ⋊ C) is finite.

For a group G and R = Z , Z / p , Q we denote by G ˆ R the R-completion of G. We study the map H n ( G , K ) → H n ( G ˆ R , K ) , where ( R , K ) = ( Z , Z / p ) , ( Z / p , Z / p ) , ( Q , Q ) . We prove that H 2 ( G , K ) → H 2 ( G ˆ R , K ) is an epimorphism for a finitely generated solvable group G of finite Prufer rank. In particular, Bousfield's HK-localisation of such groups coincides with the K-completion for K = Z / p , Q . Moreover, we prove that H n ( G , K ) → H n ( G ˆ R , K ) is an epimorphism for any n if G is a finitely presented group of the form G = M ⋊ C , where C is the infinite cyclic group and M is a C-module.

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of a finitely generated parafree augmented algebra of infinite cohomological dimension . Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and give a sufficient condition for an augmented algebra to be residually nilpotent in terms of its relations.

Let G be a group and R , S , T its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups ‖ R , S , T ‖ as well as the natural extension of the symmetric product ‖ r , s , t ‖ for corresponding ideals r , s , t in the integral group ring Z [ G ] . In this paper, it is shown that the generalized dimension subgroup G ∩ ( 1 + ‖ r , s , t ‖ ) has exponent 2 modulo ‖ R , S , T ‖ . The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for R = Q or R = Z / n and a finitely presented metabelian group G, the natural map from G to its R-completion induces an epimorphism of homology groups H 2 ( − , R ) . This answers a problem of A.K. Bousfield for the class of metabelian groups.

We give a simple proof of the Curtis' theorem: if $A_\bullet$ is $k$-connected free simplicial abelian group, then $L^n(A_\bullet)$ is an $k+ \lceil \log_2 n \rceil$-connected simplicial abelian group, where $L^n$ is the functor of $n$-th Lie power. In the proof we do not use Curtis' decomposition of Lie powers. Instead of this we use the Chevalley-Eilenberg complex for the free Lie algebra.

We introduce several classes of localizations (idempotent monads) on the category of groups and study their properties and relations. The most interesting class for us is the class of localizations which coincide with their zero derived functors. We call them right exact (in the sense of Keune). We prove that a right exact localization $L$ preserves the class of nilpotent groups and that for a finite $p$-group $G$ the map $G\to LG$ is an epimorphism. We also prove that some examples of localizations (Baumslag's $P$-localization with respect to a set of primes $P,$ Bousfield's $HR$-localization, Levine's localization, Levine-Cha's $\mathbb Z$-localization) are right exact. At the end of the paper we discuss a conjecture of Farjoun about Nikolov-Segal maps and prove a very special case of this conjecture.

Let k be an algebraically closed field of characteristic two and let Q 8 be the quaternion group of order 8. We determine the Gerstenhaber Lie algebra structure and the Batalin–Vilkovisky structure on the Hochschild cohomology ring of the group algebra k Q 8 .

We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb Q)$ is an uncountable $\mathbb Q$-vector space. Here $\hat F_\mathbb Q$ is the $\mathbb Q$-completion of $F$. This answers a problem of A.K. Bousfield for the case of rational coefficients. As a direct consequence of this result it follows that, a wedge of circles is $\mathbb Q$-bad in the sense of Bousfield-Kan. The same methods as used in the proof of the above results allow to show that, the homology $H_2(\hat F_\mathbb Z,\mathbb Z)$ is not divisible group, where $\hat F_\mathbb Z$ is the integral pronilpotent completion of $F$.

For a prime $p$, let $\hat{F}_{p}$ be a finitely generated free pro-$p$-group of rank at least $2$. We show that the second discrete homology group $H_{2}(\hat{F}_{p},\mathbb{Z}/p)$ is an uncountable $\mathbb{Z}/p$-vector space. This answers a problem of A. K. Bousfield.

In the present paper, a class of algebras that allows the so-called DTI-family of relations is introduced. With few exceptions, the stable Calabi-Yau dimension of these algebras is equal to 3. We prove that all algebras of quaternion type are contained in this class, and we give some other examples of such algebras. Furthermore, minimal projective bimodule resolutions for algebras from this class are described. Bibliography: 12 titles.

In the present paper, analogs of the Eilenberg-Watts theorem are proved for categories of finitely generated modules over finite-dimensional algebras for right exact and left exact functors. Furthermore, for left exact functors the corresponding bimodules are described explicitly. The main aim of this paper is to present how, with these versions of the Eilenberg-Watts theorem, we obtain some new descriptions of the Nakayama functor and the inverse Nakayama functor in the case of self-injective algebras. Bibliography: 4 titles.

A criterion for the sequence obtained with the help of Happel’s lemma to be a minimal bimodule projective resolution is proved. This criterion is used to construct a bimodule resolution for a family of self-injective algebras of tree class D 4. Bibliography: 6 titles.

In the present paper, the Hochschild cohomology is investigated for algebras of quaternion type from the family \( Q(2\mathcal{B})_1 \) over an algebraically closed field of characteristic 2. A 4-periodic bimodule resolution is constructed for the most part of algebras in this family. Using this resolution, a description in terms of generators and relations is given for the Hochschild cohomology algebra of the algebras under consideration. Bibliography: 20 titles.

A natural composition $\odot$ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for $p$-lower central series spectral sequence of a simplicial group. It is proved that $r$th differential satisfies a "Leibiz rule with suspension": $d^r(a\odot \sigma b)=\pm d^ra\odot b+a\odot d^r\sigma b,$ where $\sigma$ is the suspension homomorphism., Comment: 15 pages

A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop technique for the description of the higher limits over categories of presentations and show that certain homological functors can be described in this way. In particular, we give a description of Hochschild homology and the derived functors of tensor, symmetric and exterior powers in the sense of Dold and Puppe as higher limits., Comment: 25 pages

We give an equivalent definition of the stable Calabi--Yau dimension in terms of bimodule syzygies and so-called stably inner automorphisms. Using it, we complete the computation of the stable Calabi--Yau dimensions of the self-injective algebras of finite representation type which was started by K. Erdmann, A. Skowro\'nski, J. Bia\lkowski and A. Dugas., Comment: 27 pages