Cohomology of algebras

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The groups

(see Functor ), where is an associative algebra over a commutative ring with a fixed -algebra homomorphism (augmentation) enabling one to regard as an -module, and where is an -module. This definition encompasses many cohomology theories of certain types of (universal) algebras.

Cohomology groups of groups in all dimensions were introduced in the 1940s first by S. Eilenberg and S. MacLane [3] in connection with topological investigations, and by D.K. Faddeev [5] from a purely algebraic point of view — as groups of classes of generalized quotient systems. Cohomology groups in small dimensions were studied earlier in one form or another (see [1], [2], [4]).

Examples of cohomology groups.

1) If is the ring of integers, is a group, is the group ring of over , and

then the groups are called the cohomology groups of the group with coefficients (or values) in the -module ; they are denoted by . Instead of a group one can consider a monoid , and obtain the analogous cohomology groups of the monoid .

2) If is an associative -algebra, is the opposite -algebra and

then the groups are called the cohomology groups of the associative algebra with coefficients in the -bimodule (that is, in the -module ); they are denoted by . If is a field, then the groups are called the Hochschild cohomology groups of the -algebra .

3) If is a Lie algebra over a field and is its universal enveloping algebra with augmentation , then the groups are called the cohomology groups of the Lie algebra with coefficients in the -module (that is, in the Lie -module ); they are denoted by .

The cohomology groups for and 2 have, in a number of cases, simple interpretations.

a) If is a group, then is isomorphic to the group

of fixed elements; is isomorphic to the quotient group , where

is the group of derivations (or crossed homomorphisms),

is the group of inner derivations (or principal crossed homomorphisms); here, the sequence

is exact; for an Abelian group , is isomorphic to the group of extensions of by (see Baer multiplication); the third cohomology group of is connected with obstructions to extensions (see [9], Chapt. IV).

b) If is an associative -algebra, then is isomorphic to the group

is isomorphic to the quotient group

where

describes the extensions of the -bimodule by the ring (see [14]).

c) If is a Lie algebra, then is isomorphic to the -module ; is isomorphic to the quotient group

where

the second cohomology group of a Lie algebra corresponds to the -split extensions of Lie algebras (see [6], Chapt. XIV); in certain cases the elements of are obstructions in the extension problem.

Cohomology groups find extensive application in various branches of algebra. E.g. if is a group and for all -modules , then is free (Stalling's theorem, see Homological dimension). If is a finite group and is the multiplicative group of the complex field, then the group is called the Schur multiplier of . It plays an important role in the study of central extensions of groups and in the theory of projective representations of finite groups [1]. If is a group, a -module and for a prime number , then

where is the field of elements. If is a finite -group, then is the minimum number of generators of , and is the minimum number of defining relations for considered as a pro--group; , where is the minimum number of defining relations of the discrete group . The fact that tends to infinity as leads to a negative solution of the class field tower problem (cf. Class field theory), the Kurosh problem on nil algebras (cf. Nil algebra) and the unrestricted Burnside problem [10].

If is a profinite group and is the family of its open normal subgroups, then the group

is called the -th cohomology group of the profinite group with coefficients in the -module ; it is denoted by . If is a Galois extension of a field with Galois group , then is profinite; in this case the groups are called Galois cohomology groups. An important role is played by the groups where is the multiplicative group of . E.g. , and a corollary of this fact is Hilbert's theorem 90 (on cyclic extensions). If is the separable closure of , then is called the Brauer group of the field (see Brauer group). At present (1987) a Galois theory of commutative rings is being developed in which an essential part is played by the Galois cohomology of commutative rings and by the Brauer group.

If is an associative algebra, then is rigid if (see Deformation of an algebra).

In a sense, the cohomology groups are dual to the homology groups

of the associative -algebra with coefficients in an -module . If is a group, and , then the groups are called the homology groups of the group with coefficients in the -module ; they are denoted by . If is an associative -algebra and , then the groups are called the homology groups of the associative algebra with coefficients in the -bimodule ; they are denoted by . If is a Lie algebra and is its universal enveloping algebra, the groups are called the homology groups of the Lie algebra with coefficients in the Lie -module ; they are denoted by . In a number of cases, the homology groups in small dimensions have a simple interpretation. Thus, if is a group, then and .

If in an Abelian category the functor has derived functor , and the functor together with its derived functor are also defined, then the above scheme defines a cohomology and homology theory in this category. A very general approach to the construction of cohomology theories can be developed using co-triples [11]. The concept of a (co-)triple arose in the analysis of the minimal tools that are necessary for the construction of simplicial resolutions. A triple in a category is a functor together with two natural transformations of functors , , subject to the conditions

The concept of a co-triple is dual to this, that is, it is obtained by reversing arrows. If an object and a morphism are such that and , then the pair is called a -algebra. Let be the category of -algebras. If , then . This defines a functor (in a sense, is a free object over ). Let be the functor that forgets the -structure. Then and are adjoint functors (cf. Adjoint functor), , and , together with , , defines a co-triple and a complex

with differentiation (this complex is an analogue of the canonical free resolution of the object ). If is an Abelian category and the complex so obtained is acyclic, the standard application of the functor (or ) gives rise to the construction of the cohomology groups (or homology groups) of the object . In general it is necessary to construct a new Abelian category of -modules over the -algebra , on which there is a natural co-triple structure enabling one to construct groups, which are then called the cohomology groups of the original category (analogous to the construction of cohomology groups for the categories of groups, associative algebras and Lie algebras). This scheme embraces the cohomology of groups, associative algebras and Lie algebras, as well as a number of other cohomology theories (Harrison cohomology of commutative algebras, André–Quillen cohomology, Amitsur cohomology, etc; see [8]).

All the constructions specified here relate to some Abelian category. At the same time, a number of mathematical disciplines (for example, the theory of group extensions) require the construction of cohomology theories with coefficients in a non-Abelian category (for example, in a non-Abelian -module in the case of a group ) (see [8], [11]). The starting-point for the construction of various non-Abelian cohomology theories of algebras is the interpretation of cohomology in dimension 0 and 1, but certain aspects of the classical theory have to be relinquished (group structures on cohomology, etc.). Cohomology of topological algebraic structures has been considered (for example, the cohomology of topological groups [5], Banach algebras, etc.).

References

 [1] J. Schur, "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" J. Reine Angew. Math. , 132 (1907) pp. 85–137 [2] R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" Math. Z. , 38 (1934) pp. 374–416 [3] S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups" Proc. Nat. Acad. Sci. USA , 29 (1943) pp. 155–158 [4] H. Hopf, "Ueber die Bettischen Gruppen, die zu einer beliebigen Gruppe gehören" Comment. Math. Helv. , 17 (1944–1945) pp. 39–79 [5] D.K. Faddeev, "On quotient systems in Abelian groups with operators" Dokl. Akad. Nauk SSSR , 58 : 3 (1947) pp. 361–364 (In Russian) [6] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) [7] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221 [8] Itogi Nauk. Algebra 1964 (1966) pp. 203–235 [9] S. MacLane, "Homology" , Springer (1963) [10] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) [11] B. Eckmann (ed.) , Seminar on triples and categorical homology theory Zürich, 1966–1967 , Lect. notes in math. , 80 , Springer (1969) [12] K.W. Gruenberg, "Cohomological topics in group theory" , Springer (1970) [13] U. Stammbach, "Homology in group theory" , Springer (1973) [14] R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975)