Limits, standard complexes and fr-codes

TL;DR

This paper introduces a cosimplicial resolution for computing higher derived functors of limits in certain categories, with applications to K"unneth theorems and fr-code finiteness, advancing the understanding of categorical limits.

Contribution

It develops a new cosimplicial object for strongly connected categories with coproducts, enabling computation of higher limits and exploring fr-code properties.

Findings

Established a cosimplicial resolution for higher limits.

Applied the resolution to K"unneth theorem for higher limits.

Proved lim-finiteness of fr-codes for words of length ≤ 3.

Abstract

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.

Equations

$$\mathbf{f}(F\twoheadrightarrow G)=\mathsf{ker}\{\mathbb{Z}[F]\to \mathbb{Z}\},\mathbf{r}(F\twoheadrightarrow G)=\mathsf{ker}\{\mathbb{Z}[F]\to \mathbb{Z}[G]\}$$

$$\mathbf{fr}+\mathbf{rf},{\mathbf{r}}^2\cap {\mathbf{f}}^3$$

$${\mathsf{lim}}^1(\mathbf{rff}+\mathbf{frr})=\mathsf{Tor}(H_2(G),G_{ab}),$$

$${\mathsf{lim}}^1(\mathbf{rr}+\mathbf{frf}+\mathbf{rff})=H_2(G,G_{ab}),$$

$${\mathsf{lim}}^1(\mathbf{rr}+\mathbf{frf})=H_3(G),$$

$${\mathsf{lim}}^2(\mathbf{rr}+\mathbf{frf})=\mathbf{g}{\otimes }_{\mathbb{Z}[G]}\mathbf{g}.$$

$${\mathsf{lim}}^1\mathbf{rr}+\mathbf{fff}={\mathsf{lim}}^0\frac{\mathbf{f}}{\mathbf{rr}+\mathbf{fff}}=\frac{(\mathbf{rf}+\mathbf{fff})\cap (\mathbf{fr}+\mathbf{fff})}{\mathbf{rr}+\mathbf{fff}}=\mathsf{Tor}(G_{ab},G_{ab}),$$

$${\mathsf{lim}}^1\mathbf{rf}+\mathbf{fr}={\mathsf{lim}}^0\frac{\mathbf{f}}{\mathbf{rf}+\mathbf{fr}}=\frac{\mathbf{ff}}{\mathbf{rf}+\mathbf{fr}}=\mathbf{g}{\otimes }_{\mathbb{Z}[G]}\mathbf{g}.$$

$$\mathbf{rr}+\mathbf{frf}⇝\mathbf{rr}+\mathbf{frf}+\mathbf{rff}$$

$$H_3(G)={\mathsf{lim}}^1(\mathbf{rr}+\mathbf{frf})⇝H_2(G,G_{ab})={\mathsf{lim}}^1(\mathbf{rr}+\mathbf{frf}+\mathbf{frr}).$$

$${\mathrm{Hom}}_{\mathcal{C}}(c,c^{\prime })\ne \varnothing .$$

$${\mathsf{lim}}^i\mathcal{F}={\mathbb{R}}^i\mathsf{lim}\mathcal{F},\mathsf{lim}:{\mathrm{Ab}}^{\mathcal{C}}\to \mathrm{Ab}.$$

$$E_1^{p,q}={\mathsf{lim}}^q{\mathcal{F}}^p\Rightarrow \mathsf{lim}{H}^{p+q}({\mathcal{F}}^{\bullet })$$

$$\mathsf{inv}\mathcal{F}(c)=\{x\in \mathcal{F}(c)|\forall c^{\prime }\in \mathcal{C},\phi ,\psi :c\to c^{\prime },\mathcal{F}(\phi )(x)=\mathcal{F}(\psi )(x)\}.$$

$$\mathsf{lim}\mathcal{F}\cong \mathrm{eq}(\mathcal{F}(c)\rightrightarrows \mathcal{F}(c\bigsqcup c)))$$

$$\text{reflectboxB}(c{)}^n=\underset{j=0}{\overset{n}{⨆}}c,$$

$$\text{reflectboxB}(c)([n]\stackrel{f}{\to }[m])=\underset{j=0}{\overset{n}{⨆}}c\stackrel{(i_{f(0)},\dots ,i_{f(n)})}{\to }\underset{k=0}{\overset{m}{⨆}}c$$

$$d^j:c^{\bigsqcup n+1}\to c^{\bigsqcup n+2},s^j:c^{\bigsqcup n+1}\to c^{\bigsqcup n}$$

$$d^j=(i_0,\dots ,{\hat{i}}_j,\dots ,i_{n+1}),s^j=(i_0,\dots ,i_j,i_j,\dots ,i_n).$$

$$k^0{d}^0=g,k^n{d}^{n+1}=f$$

$$k^j{d}^i=\{\begin{array}{ll}{d}^i{k}^{j-1},& i<j\\ k^{j-1}{d}^j,& i=j>0\\ d^{i-1}{k}^j,& i>j+1\end{array}$$

$$k^j{s}^i=\{\begin{array}{ll}{s}^i{k}^{j+1},& i\le j\\ s^{i-1}{k}^j,& i>j\end{array}$$

$$(QA{)}^n=\mathsf{coker}\{\underset{i=1}{\overset{n}{⨁}}{A}^{n-1}\stackrel{d^i}{\to }{A}^n\}$$

$$C{A}^n=A^n,d=\sum _{i=0}^{n+1}(-1{)}^i{d}^i$$

$$k=\sum _{i=0}^n(-1{)}^i{k}^i$$

$$(\mathrm{Dec}A{)}^n=A^{n+1},d_{\mathrm{Dec}A}^i=d_A^{i+1},s_{\mathrm{Dec}A}^j=s_A^{j+1}.$$

$$(QA{)}^n=\mathsf{coker}\{(QA{)}^{n-1}\stackrel{d^1}{\to }(Q\mathrm{Dec}A{)}^{n-1}\}.$$

$${⨁}_{i=1}^{n-1}{A}^{n-2}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$k^i=(i_{0}f,\dots ,i_{i}f,i_{i}g,\dots ,i_{n}g)=s^i{\alpha }^i,{\alpha }^i:=\underset{i+1}{\underbrace{f\bigsqcup \cdots \bigsqcup f}}\bigsqcup \underset{n+1-i}{\underbrace{g\bigsqcup \cdots \bigsqcup g}}$$

$${\alpha }^j{d}^i=(i_{0}f,\dots ,i_{j}f,i_{j+1}g,\dots ,i_{n+1}g)(i_0,\dots ,{\hat{i}}_i,\dots ,i_{n+1})=(i_{0}f,\dots ,\widehat{i_{i}f},\dots ,i_{j}f,i_{j+1}g,\dots ,i_{n+1}g)$$

$$=(i_0,\dots ,{\hat{i}}_i,\dots ,i_{n+1})(i_{0}f,\dots ,i_{j-1}f,i_{j}g,\dots ,i_{n}g)=d^i{\alpha }^{j-1}$$

$${\alpha }^j{d}^i=(i_{0}f,\dots ,i_{j}f,i_{j+1}g,\dots ,\widehat{i_{i}g},\dots ,i_{n+1}g)=d^i{\alpha }^j$$

$${\alpha }^j{s}^i=(i_{0}f,\dots ,i_{j}f,i_{j+1}g,\dots ,i_{n+1}g)(i_0,\dots i_i,i_i,\dots ,i_n)$$

$$=(i_{0}f,\dots ,i_{i}f,i_{i}f,\dots ,i_{j}f,i_{j+1}g,\dots ,i_{n+1}g)=s^i{\alpha }^{j+1}$$

Full text

The authors are supported by the Russian Science Foundation grant N 16-11-10073.

1. Introduction

Let $G$ be a group. By $\mathrm{Pres}(G)$ we denote the category of presentations of $G$ with objects being free groups $F$ together with epimorphisms to $G$. Morphisms are group homomorphisms over $G$. For a functor $\mathcal{F}:\mathrm{Pres}(G)\to\mathrm{Ab}$ from the category $\mathrm{Pres}(G)$ to the category of abelian groups, one can consider the (higher) limits ${\sf lim}^{i}\,\mathcal{F},\ i\geq 0,$ over the category of presentations. The limits ${\sf lim}^{i}\,\mathcal{F}$ are studied in the series of papers [5], [6], [11], [13], [12].

Let $\mathrm{Ring}$ be the category of rings. The group ring functor $\mathbb{Z}[-]:\mathrm{Pres}(G)\to\mathrm{Ring},\ (F\twoheadrightarrow G)\mapsto\mathbb{Z}[F]$ has two functorial ideals $\mathbf{f}$ and $\mathbf{r}$ defined as

[TABLE]

For different products of ideals $\mathbf{f}$ and $\mathbf{r}$, their sums and intersections, like

[TABLE]

one can consider their higher limits. It turns out, such limits, which depend functorially on $G$, cover a rich collection of various functors on the category of groups, including certain homological functors, derived functors etc.

(Finite) sums of monomials formed from letters $\mathbf{f}$ and $\mathbf{r}$ we call ${\bf fr}$-sentences or ${\bf fr}$-codes. By translation we mean a description of the functors ${\sf lim}^{i}\left({\bf fr}-\mathrm{code}\right)$, $i\geq 1$, ${\bf fr-}\mathrm{codes}$ viewed as functors $\mathrm{Pres}(G)\to\mathrm{Ab}$. For the moment we do not have a unified method of translation of a given code and, in every new case, in order to translate a code, we find specific tricks. At the end of the paper we present a dictionary of all nontrivial translations of codes with monomials of length $\leq 3$. In order to illustrate the diversity of functors which appear in this way, we give the following examples:

[TABLE]

Here $H_{i}(G)$ is the $i$th integral homology of $G$, ${\bf g}$ the augmentation ideal in $\mathbb{Z}[G]$, $G_{ab}$ the abelianization of $G$.

Since the category $\mathrm{Pres}(G)$ is strongly connected, the ${\sf lim}^{1}({\bf fr}\mathrm{-code})$ has a natural interpretation as the maximal constant subfunctor of $\mathbf{f}/({\bf fr}\mathrm{-code})$ (see [5], [6]). For example,

[TABLE]

The point of this theory (which we also call metaphorically as ${\bf fr}$-language), is that the formal manipulations with codes in two letters may induce deep and unexpected transformations of functors. Simple transformations of ${\bf fr}$-codes, like changing the symbol $\bf r$ by $\bf f$ in a certain place, adding a monomial to the ${\bf fr}$-code etc, induce natural transformations of (higher) limits determined by these ${\bf fr}$-codes. For example, the transformation of the ${\bf fr}$-codes

[TABLE]

induces the natural transformation of functors

[TABLE]

Here the map $H_{3}(G)\to H_{2}(G,G_{ab})$ is constructed as $H_{3}(G)=H_{2}(G,{\bf g})\to H_{2}(G,{\bf g}/{\bf g}^{2})=H_{2}(G,G_{ab}),$ where the last map is induced by the natural projection ${\bf g}\twoheadrightarrow{\bf g}/{\bf g}^{2}=G_{ab}.$

This paper has two main parts. The first part is more abstract, we prove that any (finite) $\mathbf{f}\mathbf{r}$-code has only finite number of non-zero higher limits (see Theorem 4.4). In order to prove this statement, we develop a general theory of standard complexes constructed for elements of categories with pairwise coproducts (such as our category $\mathrm{Pres}(G)$). More precisely, for any object $c$ of a category with pairwise coproducts we introduce a cosimplicial object $\text{\,\sf\reflectbox{B}}(c),$ such that, for any functor $\mathcal{F}$ from our category to abelian groups, the (higher) limits ${\sf lim}^{i}\mathcal{F}$ are naturally isomorphic to the cohomotopy groups $\pi^{i}\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$ (Theorem 2.12). It follows from Theorem 4.4 that, given a ${\bf fr}$-code, the number of its non-zero higher limits is finite. In the second part we present concrete translations. We form a dictionary of the various $\mathbf{f}\mathbf{r}$-codes using spectral sequences, Grünberg resolution, Künneth-type formulas and collections of tricks. Observe that, not all $\mathbf{f}\mathbf{r}$-codes can be translated using homological algebra only, in some cases (like the case $\bf rr+ffr+frf+rff$), nontrivial statements from the theory of groups and group rings are useful.

2. The standard complex

Definition 2.1**.**

A category $\mathcal{C}$ is called strongly-connected if for any two objects $c,c^{\prime}\in\mathcal{C}$

[TABLE]

Moreover, if for any $c,c^{\prime}\in\mathcal{C}$ there exists coproduct $c\sqcup c^{\prime}$, we say that $\mathcal{C}$ is a category with pairwise coproducts (i.e. with finite non-empty coproducts).

Definition 2.2**.**

Let $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ be a functor. The Higher limits ${\sf lim}^{i}\,\mathcal{F}$ of $\mathcal{F}$ are the right derived functors of the limit functor :

[TABLE]

We will assume that in the functor category there are enough injective objects, so higher limits of any functor exists, provided $\mathcal{C}$ is small. In a general case, as in section (4), where $\mathcal{C}=\mathrm{Pres}(G)$, the existence of higher limits for functors of interest can be established, using Grothendieck-Tarsky theory, as in [6].

For a cochain complex of functors the relation between higher limits of its terms and limits of its cohomology is given by the following spectral sequence.

Proposition 2.3** ([5], (2.5), (2.6)).**

Let $\mathcal{F}^{\bullet}$ be a bounded below cochain complex of functors with ${\sf lim}$-acyclic cohomology. Then there exists a convergent spectral sequence

[TABLE]

with the differential on the first page induced by the differential of $\mathcal{F}^{\bullet}$.

Remark**.**

For a functor $\mathcal{F}$ consider its subfunctor of the invariants ${\sf inv}\,\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$:

[TABLE]

In strongly connected categories this functor is constant and its value is equal to ${\sf lim}\,\mathcal{F}$, see (4.1) in [6]. Moreover, it is known [11] that the limit of a functor from a strongly connected category with pair-wise coproducts is equal to the equalizer

[TABLE]

for any $c\in\mathcal{C}$. In particular, this equalizer does not depend on $c$.

To generalize the relation between limits and invariants to the level of derived functors we introduce the following notion:

Definition 2.4**.**

For $c\in\mathcal{C}$ consider the following cosimplicial object $\text{\,\sf\reflectbox{B}}:\Delta\to\mathcal{C}$, which we will call the standard complex associated with $c$:

[TABLE]

[TABLE]

here $i_{j}:c\to\bigsqcup_{k=0}^{m}c,\,0\leq j\leq m$ are canonical inclusions and notation $(g_{0},\dots,g_{n}),\ g_{k}:c\to c^{\prime}$ stands for the unique map $c^{\sqcup\,n+1}\to c^{\prime}$ such that $(g_{0},\dots,g_{n})\circ i_{j}=g_{j}$.

By definition, cofaces and codegeneracies of $\text{\,\sf\reflectbox{B}}(c)$

[TABLE]

are given by

[TABLE]

This complex is very similar to the so-called canonical resolution, associated with the monad $(c\sqcup(-),\nabla,i_{2})$, here $\nabla:c\sqcup c\sqcup(-)\xrightarrow{(i_{1},i_{2})\sqcup\mathrm{id}}c\sqcup(-)$, see [17] (8.6.8). This similarity will become an identification, if there is an initial object [math] in $\mathcal{C}$. In this case though all higher limits of the functor$\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ are trivial, provided $\mathcal{F}(0)=0$. Alternatively, since $(\mathcal{C},\sqcup)$ can be considered as a strong monoidal category (without unit) and every object is a monoid with respect to this structure, for any $c$ the standard resolution $\text{\,\sf\reflectbox{B}}(c)$ can be considered as a unique monoidal functor $\Delta\to\mathcal{C}$ which sends $[0]$ to $c$, as in (7.5) of [9].

Now we will study some homotopical properties of the standard complex $\text{\,\sf\reflectbox{B}}(c)$.

Definition 2.5** ([10], (2.1)).**

Let $f,g:X\to Y$ be two morphisms between cosimplicial objects $X$ and $Y$. A cosimplicial homotopy between $f$ and $g$ is a collection of maps $k^{i}:X^{n+1}\to Y^{n},\ 0\leq i\leq n$, satisfying the following identities:

[TABLE]

We will use the following definition of the Moore complex and the alternate sum complex for the abelian case, which are dualizations of the standard definitions, as in [3]:

Definition 2.6**.**

Let $A$ be a cosimplicial object in an abelian category $\mathcal{C}$

• •

The Moore complex $QA$ of $A$ is a cochain complex

[TABLE]

• •

The alternate sum complex $CA$ of $A$ is a cochain complex

[TABLE]

Both constructions are functorial, with $Q:\mathcal{C}^{\Delta}\to\mathrm{Ch}_{\leq 0}(\mathcal{C})$ being an exact functor, and as in the simplicial case, these two complexes are chain homotopic to each other. Since a cosimplicial homotopy $\{k^{i}\}_{i=0}^{\infty}$ between $f$ and $g$ induces a chain homotopy

[TABLE]

between $Cf$ and $Cg$, $Qf$ and $Qg$ are also homotopic.

The Moore complex $QA$ also has a convenient iterative description in terms of the d’ecalage of $A$, which is a cosimplcial object $\mathrm{Dec}\,A$ with the following structure:

[TABLE]

Proposition 2.7**.**

The following formula holds:

[TABLE]

Proof.

Consider the following diagram:

[TABLE]

The diagonal arrow here represents a map to a “total” cokernel of the square (the cokernel of the natural map from the push-out to the right-bottom corner), which is equal to a “sequential” cokernel, represented by the rightmost vertical arrows. ∎

Turns out, on a strongly connected $\mathcal{C}$ the standard complex construction is constant up to homotopy:

Theorem 2.8**.**

Let $\mathcal{C}$ be a category with pair-wise coproducts. Then for any two maps $f,g:c\to c^{\prime}$ the induced morphisms $\text{\,\sf\reflectbox{B}}(c)\to\text{\,\sf\reflectbox{B}}(c^{\prime})$ are homotopic.

Proof.

Consider the following collection of maps $\{k^{i}:\text{\,\sf\reflectbox{B}}(c)^{n+1}\to\text{\,\sf\reflectbox{B}}(c^{\prime})^{n}\}_{i=0}^{\infty}$:

[TABLE]

First we consider how $\alpha^{j}$ commutes with cofaces and codegeneracies. For fixed $i<j$:

[TABLE]

For $i>j+1$:

[TABLE]

For codegeneracies if $i\leq j$:

[TABLE]

And similarly for $i>j$:

[TABLE]

Returning to $k^{i}$ and using the cosimplicial identities :

[TABLE]

Finally we consider relations for $k^{j}d^{j}$ and the boundaries of the homotopy $k^{0}d^{0},\ k^{n}d^{n+1}$:

[TABLE]

This shows that $\{k^{i}\}_{i=0}^{\infty}$ defined above is indeed a cosimplicial homotopy between $\text{\,\sf\reflectbox{B}}(f)$ and $\text{\,\sf\reflectbox{B}}(g)$. ∎

Corollary 2.9**.**

Let $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ be a functor on a strongly connected $\mathcal{C}$ with pair-wise coproducts. Then the cohomology groups

[TABLE]

are independent of $c\in\mathcal{C}$.

Remark**.**

If a category $\mathcal{C}$ is not strongly connected,  

B

can be quite far from being homotopically constant, as the following example shows (see [1], [16]). Let $k$ be a field and $\mathcal{C}=k\mathrm{-Alg}$ be a category of commutative $k$-algebras and a coproduct is given by a tensor product over $k$. Let $\mathcal{F}=U:k\mathrm{-Alg}\to k\mathrm{-Mod}$ be a forgetful functor, then for $A\in k\mathrm{-Alg}$ the (coaugmented) alternate sum complex

[TABLE]

is called the Amitsur complex and its cohomology broadly depends on $A$. For example, for $A=k$, $U\text{\,\sf\reflectbox{B}}(k)=k$ and the complex is contractible. But for $A$ being a finite dimensional extension of $k$ it can be shown (see [1]) that $H^{2}(U\text{\,\sf\reflectbox{B}}(A))$ is the Brauer group of the corresponding extension.

Let $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ be a functor. Below we will study cocycles and (co)homotopy groups of the cosimplicial object $\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$.

Lemma 2.10**.**

Cofaces (4) induce isomorphisms on higher limits of $\mathcal{F}$:

[TABLE]

Proof.

First two cofaces $i_{1},i_{2}:c\to c\sqcup c$ in $\text{\,\sf\reflectbox{B}}(c)$ are inducing isomorphisms

[TABLE]

by (3.6) in [5]. Modifying the proof of this lemma, one can see that the similar fact holds for all canonical inclusions $i_{k}:c\to\sqcup^{n}c$. This can be seen by considering a functor $\Phi_{n}:c\mapsto\sqcup^{n}c$ together with a natural transformation $i_{k}:\mathrm{id}\to\Phi_{n}$ such that for any $c^{\prime}\in\mathcal{C}$ the comma category $(\Phi_{n}\downarrow c^{\prime})$ is contractible. Now consider the diagrams

[TABLE]

After applying $\mathcal{F}$ and ${\sf lim}^{n}$ diagonal arrows become isomorphisms, hence a horizontal arrow, which is a map, induced by coface, is an isomorphism too. ∎

Cocycles $\mathcal{Z}^{n}\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$ of the standard complex serve as a natural generalization of the functor of invariants (3):

Lemma 2.11**.**

For $c\in\mathcal{C}$ the following formula holds:

[TABLE]

Proof.

By definition,

[TABLE]

Let’s denote the right hand side of (9) by ${\sf inv}^{n}\,\mathcal{F}(c)$. The inclusion ${\sf inv}^{n}\,\mathcal{F}(c)\subset\mathcal{Z}^{n}\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$ is obvious. Now for any collection of maps $\varphi_{0},\dots,\varphi_{n+1}:c\to c^{\prime}$ there is a unique morphism $\Phi=(\varphi_{0},\dots,\varphi_{n+1}):\sqcup^{n+2}c\to c^{\prime}$ such that $\varphi_{j}=\Phi i_{j}$ and moreover $(\varphi_{0},\dots,\hat{\varphi}_{j},\dots,\varphi_{n+1})=\Phi\circ(i_{0},\dots,\hat{i}_{j},\dots,i_{n+1})$. Hence for $x\in\mathcal{Z}^{n}\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$:

[TABLE]

and $x\in{\sf inv}^{n}\,\mathcal{F}(c)$ ∎

The gap between the higher invariants ${\sf inv}^{n}$ and the higher limits of the functor $\mathcal{F}$ is given by the coboundaries of $\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$ as the following theorem shows and hence the standard complex (2.4) can be used as a sort of resolution for computing ${\sf lim}^{n}\,\mathcal{F}$:

Theorem 2.12**.**

For strongly connected category $\mathcal{C}$ with pair-wise coproducts and a functor $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ for any $c\in\mathcal{C}$

[TABLE]

Proof.

By (2.9) the (co)homotopy groups of $\mathcal{F}\text{\,\sf\reflectbox{B}}(c)$ are independent of $c$, in particular, a cochain complex $C\mathcal{F}\text{\,\sf\reflectbox{B}}(-)$ is bounded below, has ${\sf lim}$-acyclic cohomology and there is a spectral sequence (2.3):

[TABLE]

The first page differential in this spectral sequence (which is acting horizontally) is a morphism, induced on ${\sf lim}^{q}$ by the differential of the alternate sum complex: $\sum_{j}(-1)^{j}\mathcal{F}(d^{j})$. Each summand in this differential is an isomorphism by (2.10) and hence the first and second page of the spectral sequence look like this:

[TABLE]

Assertion follows. ∎

Definition 2.13**.**

We say that the functor $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ has the degree $\mathrm{deg}\,\mathcal{F}\leq n$ if $Q(\mathcal{F}\text{\,\sf\reflectbox{B}}(c))^{k}=0$ for all $k>n$ for some $c\in\mathcal{C}$.

This definition of the degree is a generalization (see [14]) of the usual notion of the degree of a polynomial functor between abelian categories [2]. We will sketch the (dual version of) main ideas from [14].

For a category $\mathcal{C}$ let $\mathcal{C}_{(1)}$ be a category of splitting monomorphisms of the form $c\to c\sqcup c^{\prime}$, iteratively $\mathcal{C}_{(k)}=(\mathcal{C}_{(k-1)})_{(1)}$. Given a functor $\mathcal{F}:\mathcal{C}\to\mathrm{Ab}$ its coderivative is defined as

[TABLE]

Similarly the higher orders coderivatives of $\mathcal{F}$ are defined. Then the dual version of Proposition 1.7 of [14] holds:

Proposition 2.14**.**

Let $\mathcal{F}$ be a functor such that $\mathcal{F}_{(k)}=0$ for some $k$. Then $\mathrm{deg}\,\mathcal{F}\leq k-1$.

Proof.

For a cosimplical object $X$ define $k$-cubes $c_{k}(X)$ iteratively as

[TABLE]

Then for $X=\mathcal{F}\text{\,\sf\reflectbox{B}}(c),\ \mathcal{F}_{(k-1)}(c_{k-1}(X))={\sf coker}\,\{\mathcal{F}_{(k-1)}(c_{k-1}(X))\xrightarrow{d^{1}}\mathcal{F}_{(k-1)}(c_{k-1}(\mathrm{Dec}\,X))\}$ and from (2.7) and the induction we get that

[TABLE]

. ∎

The degree functor $\mathrm{deg}$ behaves in a predictable way with a tensor product of functors :

Theorem 2.15**.**

Let $\mathcal{F}$ and $\mathcal{G}$ be functors of degrees $\leq n$ and $\leq m$ respectively. Then their tensor product $\mathcal{F}\otimes\mathcal{G}$ has degree $\leq n+m-1$.

Proof.

For a given split monomorphism $f:c\to c^{\prime}$ in $\mathcal{C}$ the map $(\mathcal{F}\otimes\mathcal{G})(f)$ divides into composition of two split monomorphisms

[TABLE]

hence coderivative of the tensor product splits as

[TABLE]

By iterating this formula we get

[TABLE]

Result now follows from this formula and Proposition 2.14. ∎

3. Künneth theorem

We can use the fact that ${\sf lim}^{n}\,\mathcal{F}$ can be expressed as cohomology groups of a well-understood complex to determine the higher limits of a tensor product of functors, using a Künneth-type spectral sequence as in (6.8) of [15]. For later use in (4) we will expand our universe of functors and describe the Künneth formula in this generalized setting.

As in [5], let $\mathrm{Mod}_{\sf r}$ denote the category of pairs $(R,M)$, where $R$ is a ring and $M$ is a right $R$-module. Morphisms are pairs $(f,\varphi):(R,M)\to(S,N)$ consisting of ring homomorphism $f:R\to S$ and $R$-linear map $\varphi:M\to N$, where $R$ acting on $N$ through $f$. There is a natural projection $\mathrm{Mod}_{\sf r}\to\mathrm{Ring}$. Similarly, $\mathrm{Mod}_{\sf l}$ will denote the category of left modules over arbitrary rings.

Definition 3.1** ([5], (3.2)).**

Let $\mathcal{O}:\mathcal{C}\to\mathrm{Ring}$ be a $\mathrm{Ring}$-valued functor. Then the right $\mathcal{O}$-module $\mathcal{F}:\mathcal{C}\to\mathrm{Mod}$ is a functor, such that the following diagram commutes:

[TABLE]

Definition of the left $\mathcal{O}$-module is completely symmetric.

Note that the higher limits ${\sf lim}^{\bullet}\,\mathcal{F}$ have a structure of a graded module over graded ring ${\sf lim}^{\bullet}\,\mathcal{O}$. For the right $\mathcal{O}$-module $\mathcal{F}$ and the left $\mathcal{O}$-module $\mathcal{G}$ their tensor product over $\mathcal{O}$ is defined as a functor:

[TABLE]

Theorem 3.2**.**

Let $\mathcal{O}:\mathcal{C}\to\mathrm{Ring}$ be a functor such that ${\sf lim}^{\bullet}\mathcal{O}$ is of finite global dimension. For a right $\mathcal{O}$-module $M$ and left $\mathcal{O}$-module $N$ of finite degree, such that $N(c)$ is a flat $\mathcal{O}(c)$-module for all $c\in\mathcal{C}$ there is a second quadrant spectral sequence

[TABLE]

Proof.

The proof is a direct combination of the cosimplicial version of Theorem 6 of [15] and (2.12). Fix $c\in\mathcal{C}$ and consider the projective resolution $P_{\bullet}$ of $M\text{\,\sf\reflectbox{B}}(c)$ over $\mathcal{O}\text{\,\sf\reflectbox{B}}(c)$ such that $\pi^{\bullet}P_{i}$ are free $\pi^{\bullet}\mathcal{O}\text{\,\sf\reflectbox{B}}(c)$ modules for all $i$. The resolution $P_{\bullet}$ can be constructed in a way that $\pi^{\bullet}P_{\bullet}$ is a free resolution of ${\sf lim}^{\bullet}\,M$ over ${\sf lim}^{\bullet}\,\mathcal{O}$ and hence there is an isomorphism of graded abelian groups

[TABLE]

∎

Applying the Moore chain complex functor $Q$ horizontally to the cosimplicial chain complex $D=P_{\bullet}\otimes_{\mathcal{O}\text{\,\sf\reflectbox{B}}(c)}N\text{\,\sf\reflectbox{B}}(c)$ and switching to the homological notation, we obtain a second quadrant double complex. Further argument is standard. Consider two spectral sequences, associated with $D$:

• •

$E^{2}_{p,q}=H_{p}^{h}H_{q}^{v}D=H_{p}Q({\sf Tor}^{\mathcal{O}\text{\,\sf\reflectbox{B}}(c)}(M\text{\,\sf\reflectbox{B}}(c),N\text{\,\sf\reflectbox{B}}(c)))$ Provided $N\text{\,\sf\reflectbox{B}}(c)$ is free as $\mathcal{O}\text{\,\sf\reflectbox{B}}(c)$-module, only the bottom line is nontrivial on the second page and the spectral sequence converges to ${\sf lim}^{\bullet}\,M\otimes_{\mathcal{O}}N$

• •

$E^{2}_{p,q}=H_{q}^{v}H_{p}^{h}D=H_{q}(\pi^{\bullet}P_{p}\otimes_{{\sf lim}^{\bullet}\,\mathcal{O}}{\sf lim}^{\bullet}\,N)=({\sf Tor}_{p}^{{\sf lim}^{\bullet}\,\mathcal{O}}({\sf lim}^{\bullet}\,M,{\sf lim}^{\bullet}\,N))_{q}$

Since ${\sf Tor}$-functors vanish above the certain line this spectral sequence converges to the same limit, as the first one.

4. $\mathbf{f}\mathbf{r}$-codes

We denote by $\mathrm{Pres}$ the category whose objects are all presentations $c:F\twoheadrightarrow G$ and morphisms are commutative squares

[TABLE]

For each group $G$ the category $\mathrm{Pres}(G)$ is a subcategory of $\mathrm{Pres}.$ Then for a functor

[TABLE]

and any $i\geq 0$ we have a map

[TABLE]

Here $c:F\twoheadrightarrow G$ can be considered as an object of $\mathrm{Pres}(G)$ and we can take $\text{\,\sf\reflectbox{B}}(c)$ that we will denote by $\text{\,\sf\reflectbox{B}}_{G}(c)$ in order to emphasize that we take it in the category $\mathrm{Pres}(G)$ but not in the whole category $\mathrm{Pres}.$ By Theorem 2.12 we have an isomorphism

[TABLE]

Moreover any morphism (11) in the category $\mathrm{Pres}$ gives a morphism of cosimplicial objects

[TABLE]

Then the morphism $(\varphi,\widetilde{\varphi})$ induces a homomorphism

[TABLE]

Lemma 4.1**.**

The homomorphism (12) depends only on $\varphi$ and does not depend on the choice of presentations and $\widetilde{\varphi}.$ Moreover, these homomorphisms define a functor

[TABLE]

Proof.

Assume that we have two presentations $c_{i}:F_{i}\twoheadrightarrow G,$ $i=1,2$ for $G,$ two presentations $c_{i}^{\prime}:F_{i}^{\prime}\twoheadrightarrow G^{\prime}$ for $G^{\prime}.$ Assume also that we have two morphisms $(\varphi,\widetilde{\varphi}_{i}):c_{i}\to c_{i}^{\prime}$ in $\mathrm{Pres}.$ Consider the presentations $c_{1}*c_{2}:F_{1}*F_{2}\twoheadrightarrow G$ and $c_{1}^{\prime}*c_{2}^{\prime}:F_{1}^{\prime}*F_{2}^{\prime}\twoheadrightarrow G^{\prime},$ the morphism $(\varphi,\widetilde{\varphi}_{1}*\widetilde{\varphi}_{2}):c_{1}*c_{2}\to c_{1}^{\prime}*c_{2}^{\prime}$ and the commutative diagram

[TABLE]

By Theorem 2.8 the vertical arrows induce isomorphisms on $H^{i}\mathcal{F}\text{\,\sf\reflectbox{B}}(-).$ The assertion follows. ∎

The group ring functor $\mathbb{Z}[-]:\mathrm{Pres}\to\mathrm{Ring},\ (F\twoheadrightarrow G)\mapsto\mathbb{Z}[F]$ has two functorial ideals ($\mathbb{Z}[F]$-modules in sense of Definition 3.1) $\mathbf{f}$ and $\mathbf{r}$ defined as

[TABLE]

Definition 4.2**.**

The $\mathbb{Z}[F]$-module ${\bf c}:\mathrm{Pres}\to\mathrm{Ab}$ is called an $\mathbf{f}\mathbf{r}$-code, if it is a functorial ideal of $\mathbb{Z}[F]$, formed by products of the ideals $\mathbf{f}$ and $\mathbf{r}$, their sums and intersections.

Usually we consider ${\bf c}$ as a functor from $\mathrm{Pres}(G)\to\mathrm{Ab}$ for a fixed $G,$ limits always are taken over $\mathrm{Pres}(G).$ We need it to be defined on the category $\mathrm{Pres}$ only for the functors

[TABLE]

to be well-defined. (see Lemma 4.1).

The notion of degree (2.13) seems to be a reasonable invariant of $\mathbf{f}\mathbf{r}$-code for the estimation of its ${\sf lim}^{\bullet}$-dimension, since the Moore chain complex functor $Q$ is exact and the property of a functor being a degree $\leq k$ is closed under extensions. But already $\mathbf{f}$ itself has an infinite degree, although it is $\mathbb{Z}[F]$-additive (i.e. of degree one with respect to $\mathbb{Z}[F]$), as shown in [6]. But since all $\mathbf{f}\mathbf{r}$-codes are subfunctors of $\mathbf{f}$, this difficulty can be overcame by introducing the following notion:

Definition 4.3**.**

An $\mathbf{f}$-degree of an $\mathbf{f}\mathbf{r}$-code $\mathbf{c}$ is a degree of the quotient $\mathbf{f}/\mathbf{c}$.

Since $\mathbf{f}$ has trivial limits, it is straightforward that if $\mathrm{deg}^{\mathbf{f}}\,\mathbf{c}\leq n$ then ${\sf lim}^{i}\,\mathbf{c}=0$ for $i>n+1$.

Theorem 4.4**.**

Every (finite) $\mathbf{f}\mathbf{r}$-code $\mathbf{c}$ has a finite $\mathbf{f}$-degree and hence only a finite number of the non-zero higher limits.

Proof.

Let $n$ be a minimal power of $\mathbf{r}$ such that $\mathbf{r}^{n}\subset\mathbf{c}$, then we have an epimorphism $\mathbf{f}/\mathbf{r}^{n}\twoheadrightarrow\mathbf{f}/\mathbf{c}$ which induces a surjection on the level of cochain complexes:

[TABLE]

and hence it is sufficient to prove finiteness of $\mathbf{f}/\mathbf{r}^{n}$. The sequence of the short exact sequences

[TABLE]

starts with a constant functor $\mathbf{f}/\mathbf{r}=\mathbf{g}={\sf ker}\,\{\mathbb{Z}[G]\to\mathbb{Z}\}$ and the problem is reduced to the functors $\mathbf{r}^{n}/\mathbf{r}^{n+1}=(\mathbf{r}/\mathbf{r}^{2})^{\otimes_{\mathbb{Z}[F]}n}$ (see Lemma 5.1). Covering this tensor product by the tensor product over $\mathbb{Z}$ and applying Theorem 2.15, only the case $n=1$ need to be shown. Note that $\mathbf{r}/\mathbf{r}^{2}$ is a free $\mathbb{Z}[G]$-module with basis formed by elements $r-1,\ r\in R$, see [4], hence a natural map $R_{ab}\to\mathbf{r}/\mathbf{r}^{2},\ r\mapsto r-1$ factors through $\mathbb{Z}[G]\otimes R_{ab}\to\mathbf{r}/\mathbf{r}^{2}$ and this map is an isomorphism.

Finally, the functor $R_{ab}=\mathbf{r}/\mathbf{f}\mathbf{r}$ has a finite degree, since it is embedded in $\mathbf{f}/\mathbf{f}\mathbf{r}=\mathbf{f}\otimes_{\mathbb{Z}[F]}\mathbb{Z}[G]$ which is an additive functor. Indeed (see also [17]):

[TABLE]

[TABLE]

which concludes the proof. ∎

5. Dictionary

In this section, we give a dictionary for all codes written on ${\bf fr}$-language which consist of words with length $\leq 3$. If one can not find a code in our table, this means that either it has trivial translation, i.e. all ${\sf lim}^{i}=0$, or has the same translation as its mirror image, which is in our dictionary. For example, the codes ${\bf rf+ffr}$ and ${\bf fr+rff}$ have the same translations. As mention in Introduction, by translation we mean a description of the functors ${\sf lim}^{i}\left({\bf fr}-\mathrm{code}\right)$, $i\geq 1$, ${\bf fr-}\mathrm{codes}$ viewed as functors from the category of free group presentations to the category of abelian groups.

We will omit the translation of simple codes given in [6], like $\bf rr+fff$, or $\bf rr+frf,\ rrf+frr$.

In construction of the dictionary, we will use the following statements.

Lemma 5.1** (Lemma 5.9 in [6]).**

Let ${\bf a^{\prime}\subset a,\ b^{\prime}\subset b}$ be ideals of $\mathbb{Z}[F]$ and ${\sf Tor}(\mathbb{Z}[F]/{\bf a},\mathbb{Z}[F]/{\bf b})=0$, then there is a natural isomorphism

[TABLE]

Lemma 5.2**.**

For any functor $\mathcal{F}(F,R)$ and a non-constant functor $\mathcal{H}(F)$, which depends only on $F$, ${\sf lim}^{i}\mathcal{F}\otimes\mathcal{H}=0,\ i\geq 0$.

Similarly one can show (see [6]) that, for a $\bf fr$-code with all words started with $\bf f,$ all limits are zero.

Lemma 5.3** (Lemma 6 in [11]).**

Let $\mathcal{F}$ be a constant functor. Then any subfunctor $\mathcal{G}\hookrightarrow\mathcal{F}$ and any epimorphic image $\mathcal{F}\twoheadrightarrow\mathcal{H}$ are constant functors.

We will also use the spectral sequence 2.3, especially applied to the 4-term complexes. For convenience, lets reformulate the statement about convergence of the spectral sequence 2.3 in a more explicit form. Let $\mathcal{F}^{\bullet}$ be a complex of functors $\mathrm{Pres}(G)\to\mathrm{Ab}$

[TABLE]

Assume that $\mathcal{F}^{\bullet}$ is bounded below (i.e. $\mathcal{F}^{n}=0$ for $n<\!<0$) and that $H^{n}(\mathcal{F}^{\bullet})$ is constant for any $n.$ Then there exists a converging spectral sequence $E$ with differentials

[TABLE]

such that

[TABLE]

Now we proceed to the computations.

rfr+frf: Tensoring the short exact sequence $\frac{\bf r}{\bf fr}\hookrightarrow\frac{\bf f}{\bf fr}\twoheadrightarrow{\bf g}$ by $-\otimes\frac{\bf f}{\bf fr+rf}$ and taking the group homology $H_{i}(G,-)$, we get the long exact sequence

[TABLE]

Here we used the property that, for any $G$-module $M$, there is a natural isomorphism $H_{1}(G,{\bf g}\otimes M)=H_{2}(G,M)$. Since $\mathbf{f}/\mathbf{f}\mathbf{r}$ is a free $\mathbb{Z}[G]$-module, $\frac{\bf f}{\bf fr}\otimes\frac{\bf f}{\bf fr+rf}$ is weak projective and hence $H_{i}\left(G,\frac{\bf f}{\bf fr}\otimes\frac{\bf f}{\bf fr+rf}\right)=0,\ i\geq 1$. By Lemma 5.1,

[TABLE]

And $\frac{\bf ff}{\bf ffr+frf}$ has trivial limits by Lemma 5.2. From a spectral sequence of Proposition 2.3 applied to a four-term exact sequence (14), it can be seen that

[TABLE]

and there is a short exact sequence

[TABLE]

Here $\mathbf{g}\otimes_{\mathbb{Z}[G]}\mathbf{g}^{2}={\sf lim}(\mathbf{g}\otimes_{\mathbb{Z}[G]}\frac{\mathbf{f}}{\mathbf{f}\mathbf{r}+\mathbf{r}\mathbf{f}})={\sf lim}^{1}({\bf rf+ffr})$. To determine the ${\sf lim}$ and ${\sf lim}^{1}$ of $H_{2}(G,\frac{\bf f}{\bf fr+rf})$, consider the short exact sequence

[TABLE]

and the associated homology long exact sequence

[TABLE]

Any map from $H_{n}(G)\otimes F_{ab}$ to a constant functor (which depends only on $G$) factors through $H_{n}(G)\otimes G_{ab}$. This follows from elementary properties of colimits (see [7]), namely from ${\sf colim}(H_{n}(G)\otimes F_{ab})=H_{n}(G)\otimes G_{ab}$. Therefore, after truncating (15) and applying Proposition 2.3 together with Lemma 5.3 to it, we obtain

[TABLE]

and

[TABLE]

Hence, there is a short exact sequence

[TABLE]

rr+frf+rff: Consider the Grünberg resolution which consists of free $\mathbb{Z}[G]$-modules:

[TABLE]

Tensoring it with $G_{ab}=\frac{\bf f}{\bf r+ff}$ over $\mathbb{Z}[G]$, we obtain the complex

[TABLE]

which can be written, by Lemma 5.1 as

[TABLE]

Hence, there is a natural isomorphism

[TABLE]

For two ideals $I,J\subset\mathbf{f}$ there is a short exact sequence:

[TABLE]

And hence we get the following 4-term exact sequence

[TABLE]

From the associated spectral sequence we obtain the identifications

[TABLE]

Now observe that, the statement written in Introduction, that the transformation of the ${\bf fr}$-codes

[TABLE]

induces the natural transformation of functors

[TABLE]

follows immediately from the identifications

[TABLE]

rrf+rfr+frr: Taking the tensor product over $\mathbb{Z}[G]$ of the Grünberg resolution with $H_{2}(G)=\frac{\bf r\cap ff}{\bf fr+rf}$ and $\frac{\bf r}{\bf fr+rf}$ respectively, we obtain

[TABLE]

Since ${\sf lim}^{1}{\bf rr}={\sf lim}^{1}({\bf ffr+frf})=0,$

[TABLE]

The natural map $H_{2}(G,H_{2}(G))\to H_{2}\left(G,\frac{\bf r}{\bf fr+rf}\right)$ is injective. Indeed, the above terms can be decomposed as

[TABLE]

and

[TABLE]

Using the fact that $\frac{\bf r}{\bf r\cap ff}={\sf coker}\{H_{2}(G)\hookrightarrow\frac{\mathbf{r}}{\mathbf{f}\mathbf{r}+\mathbf{r}\mathbf{f}}\}$ is torsion-free (since it is a subgroup of $\mathbf{f}/\mathbf{f}\mathbf{f}=F_{ab}$), we see that the natural map $H_{2}(G)\otimes H_{2}(G)\to H_{2}(G)\otimes\frac{\bf r}{\bf fr+rf}$ is injective and ${\sf Tor}(G_{ab},H_{2}(G))\to{\sf Tor}\left(G_{ab},\frac{\bf r}{\bf fr+rf}\right)$ is an isomorphism. The natural map $H_{1}(G,H_{2}(G))\to H_{1}\left(G,\frac{\bf r}{\bf fr+rf}\right)$ is also injective, by the same reason. Hence, we have the following short exact sequence

[TABLE]

It follows that

[TABLE]

To compute the latter one, we use Künneth theorem 3.2, which in this case degenerates to

[TABLE]

Applying Proposition 2.3 to the 4-term exact sequence

[TABLE]

we obtain the following description

[TABLE]

The isomorphism (16) and the exact sequence (17) now imply that there exists the following natural short exact sequence

[TABLE]

In order to understand ${\sf lim}^{2}({\bf rrf+rfr+frr}),$ consider the spectral sequence applied to the 4-term sequence

[TABLE]

Putting the values of ${\sf lim}^{i}({\bf rr+frf+ffr}),{\sf lim}^{1}H_{2}\left(G,\frac{\bf r}{\bf rf+fr}\right)$ into the cells of the spectral sequence and noting that ${\sf lim}^{2}{\bf rr}={\bf g\otimes g},\ {\sf lim}^{i}{\bf rr}=0,\ i\neq 2,$ we obtain the following diagram which gives a description of ${\sf lim}^{2}({\bf rrf+rfr+frr})$ as a functor glued from three pieces

[TABLE]

ffr+rff: First observe that,

[TABLE]

This follows from the isomorphism

[TABLE]

[TABLE]

Consider the following exact sequence

[TABLE]

The left hand term is ${\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g}^{2}$. Since an epimorphic image of a constant functor is a constant functor, (18) implies that

[TABLE]

where $\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}}{\sim}$ is the image of the left hand map in the last exact sequence, i.e.

[TABLE]

rr+ffr+rff: Define one more quotient of ${\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}$ as follows111Observe that, there exists a natural exact sequence

${\sf Tor}(G_{ab},G_{ab})\to{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}\to\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}}{\approx}\to 0.$

:

[TABLE]

There is a natural epimorphism $\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}}{\sim}\twoheadrightarrow\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g^{2}}}{\approx}.$ The short exact sequence

[TABLE]

implies that

[TABLE]

and there is an exact sequence

[TABLE]

Now consider the short exact sequence

[TABLE]

The left hand term has zero limits, since it is isomorphic to $G_{ab}\otimes F_{ab}\otimes G_{ab}$. Since the diagonal action of $F$ on the middle and the right head terms are trivial, they are isomorphic to $\frac{\bf f(ff+r)}{\bf(r+ff)^{2}}$ and $\frac{\bf f(ff+r)}{\bf rr+fff}$ respectively. Hence,

[TABLE]

We obtain the needed description

[TABLE]

and the short exact sequence

[TABLE]

frr+rfr: There is an isomorphism

[TABLE]

This is a particular case of the functor $A\otimes_{\mathbb{Z}[F]}{\bf r}$, where $A$ is a constant, in this case $A=\mathbf{g}\otimes_{\mathbb{Z}G}\mathbf{g}$. Since ${\sf lim}^{\bullet}\mathbb{Z}[F]=\mathbb{Z}$ is of finite global dimension and $\bf r$ is a free $\mathbb{Z}[F]$ -module, Künneth theorem 3.2 can be applied to $A\otimes_{\mathbb{Z}[F]}\bf r$, and it degenerates to a series of usual Künneth short exact sequences

[TABLE]

which computes the only non-trivial higher limit as

[TABLE]

In this way, we obtain the description

[TABLE]

In the same way we have

[TABLE]

rff+frr: There is an isomorphism

[TABLE]

We have the following descriptions of the limits of above terms

[TABLE]

and

[TABLE]

As noted before, the abelian group $\frac{\bf r+ff}{\bf ff}={\bf\frac{r}{r\cap ff}}$ is torsion-free, hence the Künneth formula implies the following

[TABLE]

Comparing this description with the values of ${\sf lim}^{i}({\bf rr+rff})={\sf lim}^{i}({\bf rr+ffr}),$ we obtain the following:

[TABLE]

ffr+frf+rff+rr: Consider the short exact sequence

[TABLE]

The left hand term is a natural quotient of ${\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\otimes_{\mathbb{Z}[G]}\bf g$, hence,

[TABLE]

Next observe that,

[TABLE]

This follows from the identification of the intersection of augmentation ideals222In the free group ring $\mathbb{Z}[F],$ ${\bf rr}=\Delta^{2}(R)+{\bf rrf}.$ $\Delta^{2}(R)\cap{\bf fff}=\Delta^{3}(R)+\Delta(R)\Delta(R\cap[F,F])+\Delta([R,R]\cap[[F,F],F])$ and the identity $R^{\prime}\cap\gamma_{3}(F)=[R\cap F^{\prime},R]$333A simple proof of this identity is the following. Observe that, $\Lambda^{2}(R/(R\cap[F,F]))=\frac{[R,R]}{[R,R\cap[F,F]]},$ where $\Lambda^{2}$ is the exterior square, and $\Lambda^{2}(F_{ab})=[F,F]/[[F,F],F]$. Now the needed identity follows from the inclusion $\Lambda^{2}(R/(R\cap[F,F]))\hookrightarrow\Lambda^{2}(F_{ab}),$ which is induced by the inclusion $R/R\cap[F,F]\hookrightarrow F_{ab}$., see [8]. Hence, the left hand term in the above short exact sequence is ${\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\otimes_{\mathbb{Z}[G]}\bf g$ itself and we have a short exact sequence

[TABLE]

We collect the results in the following table. By $F"\oplus"G$ we mean an extension of the form $F\hookrightarrow*\twoheadrightarrow G$.

fr-code ${\bf lim}^{1}$ ${\bf lim}^{2}$ ${\bf lim^{3}}$

r ${\bf g}$ 0 0

rr 0 ${\bf g}\otimes{\bf g}$ 0

rrr 0 0 ${\bf g}\otimes{\bf g}\otimes{\bf g}$

fr+rf ${\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}$ 0 0

ffr+frf+rff ${\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}$ 0 0

r+ff $G_{ab}$ 0 0

r+fff ${\bf g}/{\bf g}^{3}$ 0 0

rf+ffr ${\bf g}^{2}\otimes_{G}{\bf g}$ 0 0

fr+rf+fff $G_{ab}\otimes G_{ab}$ 0 0

rr+fff ${\sf Tor}(G_{ab},G_{ab})$ $G_{ab}\otimes G_{ab}$ 0

rr+frf $H_{3}(G)$ ${\bf g}\otimes_{G}{\bf g}$ 0

rrf+frr $H_{4}(G)$ $({\bf g}\otimes{\bf g}\otimes{\bf g})_{G}$ 0

rfr+frf ${\sf coker}\{H_{3}(G)\otimes G_{ab}\to$ ${\sf im}\{H_{2}(G)\otimes G_{ab}\to$ 0

$H_{2}\left(G,{\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\right)\}$ $H_{1}\left(G,{\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\right)\}"\oplus"{\bf g}^{2}\otimes_{\mathbb{Z}[G]}\bf g$

rff+ffr $\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g}^{2}}{\sim}$ 0 0

rr+frf+rff $H_{2}(G,G_{ab})$ $G_{ab}\otimes G_{ab}$ 0

rr+ffr 0 $G_{ab}\otimes{\bf g}$ 0

rfr+frr 0 $({\bf g}\otimes_{\mathbb{Z}[G]}{\bf g})\otimes{\bf g}$ 0

rr+ffr+rff $\frac{{\bf g}^{2}\otimes_{\mathbb{Z}[G]}{\bf g}^{2}}{\approx}\ "\oplus"\ {\sf Tor}(G_{ab},G_{ab})$ $G_{ab}\otimes G_{ab}$ 0

rr+ffr+frf+rff ${\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\otimes_{\mathbb{Z}[G]}{\bf g}\ "\oplus"\ {\sf Tor}(G_{ab},G_{ab})$ $G_{ab}\otimes G_{ab}$ 0

rff+frr ${\sf Tor}(H_{2}(G),G_{ab})$ $H_{2}(G)\otimes G_{ab}\ "\oplus"\ {\sf Tor}(G_{ab},G_{ab})$ 0

$"\oplus"\ {\sf ker}\{{\bf g}\otimes G_{ab}\twoheadrightarrow G_{ab}\otimes G_{ab}\}$

rrf+rfr+frr $H_{2}(G,H_{2}(G))\ "\oplus"\ {\sf Tor}(G_{ab},H_{2}(G))$ $H_{2}(G)\otimes G_{ab}\ "\oplus"\ H_{2}(G,G_{ab})$ 0

$"\oplus"\ {\sf ker}\{{\bf g\otimes g}\to G_{ab}\otimes G_{ab}\}$

nty

A.SAS.

hash=d35756c9f39a938d9533376dfaa5ad43MacLaneM.SaundersS.

hash=376eb52322c614eb9f5ed36fcb5aa886JardineJ.John FJ.F.

G.Karl WK.W.

hash=c1fbe109a0b4942a15e3ce511f393180MikhailovM.RomanR.

hash=c1fbe109a0b4942a15e3ce511f393180MikhailovM.RomanR.

hash=c1fbe109a0b4942a15e3ce511f393180MikhailovM.RomanR.hash=cb6aabee36fa58a187570d55f54dce00SosniloS.VladimirV.

hash=ac480063a097b769bae5aa33cd291d46KumarK.DeepakD.hash=d84e3282576807d21e5ce27b446d3b56VermaniV.LRL.

M.L.SaundersS.

M.Jean-PierreJ-P.

hash=68021a04e8e6dff842a5c2c3204a59b7PassiP.Inder Bir SI.B.S.

hash=68021a04e8e6dff842a5c2c3204a59b7PassiP.Inder Bir SI.B.S.

hash=68021a04e8e6dff842a5c2c3204a59b7PassiP.Inder Bir SI.B.S.

P.TT.

Q.Daniel GD.G.

hash=535b7933f0622d2d0906c9d3021fa810ZelinskyZ.DanielD.

W.Charles AC.A.