Non-Abelian Homological Algebra and Its Applications

267 Pages·1997·7.6 MB·English

by  Hvedri Inassaridze (auth.)

Non-Abelian Homological Algebra
and Its Applications
Mathematics and Its Applications
ManagingEditor:
M.HAZEWINKEL
Centrefor MathematicsandComputerScience.Amsterdam.TheNetherlands
Volume421
Non-Abelian
Homological Algebra
and Its Applications
by
Hvedri Inassaridze
GeorgianAcademyofSciences,
Tbilisi,Georgia
Springer-Science+Business Media, B.V
AC.I.P.Cataloguerecord forthis book isavailable fromtheLibrary ofCongress.
ISBN978-90-481-4899-8 ISBN978-94-015-8853-9(eBook)
DOI 10.1007/978-94-015-8853-9
Printedonacid-freepaper
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@1997 SpringerScience+BusinessMediaDordrecht
Originally publishedbyKluwerAcademic Publishers in 1997.
Softcoverreprint ofthehardcover 1stedition 1997
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Contents
Introduction 1
1 Universal sequences of functors 5
2 Derived functors 35
3 Extensions and cohomology ofmonoids with coefficients
in semimodules 79
4 Cohomology of algebraic theories and MacLane
(co)homology of rings 107
5 K-functors 163
6 Non-abelian homology of groups 191
References 257
Introduction
While in classical (abelian) homological algebra additive functors from
abelian (or additive) categories to abelian categories are investigated ,
non- abelian homological algebra deals with non-additive functors and
their homological properties , in particular with functors having values
in non-abelian categories. Such functors haveimportant applications in
algebra, algebraic topology, functional analysis, algebraic geometry and
other principal areas of mathematics. To study homological properties
of non-additive functors it is necessary to define and investigate their
derived functors and satellites. It will be the aim of this book based
on the results of researchers of A.Razmadze Mathematical Institute of
the Georgian Academy of Sciences devoted to non-abelian homological
algebra. The most important considered cases will be functors from
arbitrary categories to the category of modules, group valued functors
and commutative semigroup valued functors.
In Chapter I universal sequences of functors are defined and in
vestigated with respect to (co)presheaves of categories, extending in a
natural way the satellites of additive functors to the non-additive case
and generalizing the classical relative homological algebra in additive
categories to arbitrary categories. Applications are given in the furth
coming chapters.
Chapter II is devoted to the non-abelian derived functors of group
valued functors with respect to projective classesusing projective pseu
dosimplicial resolutions. Their functorial properties (exactness, Milnor
exact sequence, relationship with cotriple derived functors, satellites
and Grothendieck cohomology, spectral sequence of an epimorphism,
degree of an arbitrary functor) are established and applications to ho
mology and cohomology of groups are given.
1
H. Inassaridze, Non-Abelian Homological Algebra and Its Applications
© Springer Science+Business Media Dordrecht 1997
2 INTRODUCTION
In Chapter III the monoid cohomology of monoids with coefficients
in semimodules,generalizing the classical cohomology theory ofgroups,
its relationship with extensions of monoids by semimodules and its
functorial properties are treated.
Chapter IV is devoted to MacLane cohomology and homology of
rings. MacLane cohomology of rings is extended and described with
the use of extension functors in the category of all functors from the
category offreefinitely generated left modules to the category ofall left
modules. Ageneralization ofMacLane cohomologyofrings to algebraic
theories and its relationship with Barr and Beck's cotriple cohomology
are given. The isomorphism of MacLane homology of discrete rings
with topological Hochschild homology is established.
In Chapter V the algebraic and topological K-functors are obtained
with the use of left derived functors of the general linear group func
tor GL with respect to the projective class of free groups for Quillen's
K-theory, with respect to the projective class of free rings for Swan's
K-theory, with respect to the projective class of polynomial extensions
for Karoubi-Villamayor'sK-theory, with respect to the projective class
generated by the path cotriple for Atiyah's K-theory. The interpreta
tion of Kasparov's bivariant K-theory as the right satellite in the cate
gory of separable C*-algebras of the Grothendieck functor of the com
mutative monoid of homotopy classes of homomorphisms with respect
to the (co)presheaf of semi-split extensions of separable C*-algebras is
given.
In Chapter VI a non-abelian tensor product ofgroups for non com
patible actions is introduced, generalizing Brown- Loday's non-abelian
tensor product of groups. Some functorial properties (exactness, ad
junction with crossed homomorphisms) of this tensor product are es
tablished and its finiteness is investigated ( sufficient conditions for
the finiteness of the non-abelian tensor product of finite groups is ob
tained , generalizing Ellis's result for compatible actions). Using the
left derived functors of the non-abelian tensor product ofgroups in the
category of groups acting on a fixed group (which acts morphismpre
serving on each object of this category) with respect to the projective
classoffreegroups a non-abelian homologyofgroups isdefined, extend
ing the classical homology of groups to the case when the coefficients
are arbitrary groups, and application ofthis homology to the K-theory
INTRODUCTION 3
of noncommutative local rings is given. FollowingGuin's approach to
non-abelian cohomology of groups the second non-abelian cohomology
with coefficientsin crossed modules is defined and Guin's six terms ex
act cohomologysequence isextended to a nine terms exact cohomology
sequence. The notion of crossed bimodule is introduced, generalizing
in a natural way the notion of crossed module and a non-abelian coho
mology ofgroups with coefficientsin crossed bimodules is constructed,
preserving the exactness with respect to the coefficients.
Chapter 1
Universal sequences of
functors
Our aim is to extend the well-known notion of satellites of additive
functors to the non-additive case using the universal proprety ofsatel
lites.
Universal sequences ofarbitrary functors (non necessarily additive)
will be introduced [38,40,43] generalizing in a natural way the classi
cal definitions of Grothendieck [32] and MacLane [68], as well as of
Buchsbaum [12], Heller [37] and Butler-Horrocks [14].
One gets a general construction of relative homological algebra in
arbitrary categories including the relative homologicalalgebra in addi
tive categories with respect to proper classes ofshort exact sequences.
Let A be an arbitrary category and let K be the "category" whose
objects are categories and morphims are covariant functors.
Definition 1.1. It will be said that G is a contravariant functor
from A to K if G assigns to each object A of A an object G(A) of K
and to each morphism f :A' --t A of A a morphism G(J) : G(A) --t
G(A') ofK such that G(lA) is equivalent to the identity functor 1G(A)
and G(Jg) is equivalent to the composite G(J)G(g) of functors G(J)
and G(g) for f : A' --t A and 9 : A" --t A'. The contravariant
functor G will be called a presheaf of categories over the category A.
Acopresheafofcategories overthe category A isa covariant functor
H (up to equivalence) from A to K.
Let A and B be arbitrary categories and let G be a presheaf of
5
H. Inassaridze, Non-Abelian Homological Algebra and Its Applications
© Springer Science+Business Media Dordrecht 1997
6 CHAPTER 1. UNIVERSAL SEQUENCES OF FUNCTORS
categories over the category B.
Definition 1.2. It willbe said that (S, s) is a trace in the category
A of the presheaf G if for any B E Ob B a covariant functor SB :
G(B) -t A and for any object E of G(B) and every morphism f :
B' -t B of B a morphism SE(J) : SB/[G(J)(E)] -t 8B(E) of the
category A are given such that the following conditions are satisfied:
1) for any morphism a : E1 -t E2 of G(B) and any morphism
f :B/ -t B the diagram
8B/[G(J)(E1)] SB'[~G(f)(a)] 8B'[G(J)(E2)]
4- (J) 4- SE2(J)
SEl
8B(Ed 8B(E2)
is commutative;
2) for any E E Ob G(B), B E B,
sE(lB) .8B('I9)= ls
B(E)
where '19 : E -t G(lB)(E) is the isomorphism induced by the equiva
lenceof functors 1G(B) and G(lB);
3) for f : B' -t B, 9 : B" -t B' of the category B and for any
E E Ob G(B)
SE(J).SG(f)(E) = SE(Jg) ·8B,,('I9)
where '19 : G(g)[G(J)(E)] -t G(Jg)(E) is the isomorphism induced
by the equivalenceof functors G(g)G(J) and G(Jg).
Definition 1.3. It willbe said that (8, s) is a trace in the category
A of the copresheaf Hover B if for any B E Ob B a covariant functor
8B : H(B) -t A and for any E E Ob H(B) and every morphism
f :B -t B' of B a morphism SE(J) :8B(E) -t 8B,[H(J)(E)] of the
category A are given such that the following conditions are satisfied;
1) for any morphism a : E1 -t E2 of H(B) and any morphism
f :B -t B' of B the diagram
SE(a)
8B(Ed ..=...2-f 8B(E2)
4- SE1(J) 4- SE2(J)
8B;[H(J)(Ed] 8B'[H(J)(E2)]