Homological algebra

The branch of algebra whose main study is derived functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).

One of the origins of homological algebra is the (singular) homology theory of topological spaces. In this, to each topological space $ X $ is associated a sequence of abelian groups $ {H_{n}}(X) $ (homology groups), while to a continuous mapping $ f: X \to Y $ of spaces is associated a set of homomorphisms $ f_{n}: {H_{n}}(X) \to {H_{n}}(Y) $ of homology groups. Each $ n $-dimensional singular simplex $ T $ of a topological space $ X $ has a boundary consisting of singular simplices of dimension $ n - 1 $. If $ K_{n} $ is the free abelian group generated by all the $ n $-dimensional simplices, then the function $ \partial $ that assigns to each $ T $ the alternating sum $ \partial T $ of its boundary simplices defines a homomorphism $ \partial: K_{n} \to K_{n - 1} $ so that $ \cdots \stackrel{\partial}{\longrightarrow} K_{n - 1} \stackrel{\partial}{\longrightarrow} K_{n - 1} \stackrel{\partial}{\longrightarrow} K_{n - 2} \stackrel{\partial}{\longrightarrow} \cdots $ is a complex of abelian groups, and a continuous mapping of spaces induces homomorphisms of their respective complexes. Certain properties of the space $ X $ or the mappings $ f: X \to Y $ may be deduced from the properties of the homology groups $ H_{n} $ of this complex, or from the corresponding homomorphisms $ f_{n} $ of these homology groups. This makes it possible, in a number of cases, to reduce the study of topological objects to the study of certain algebraic objects, as is done in analytic geometry (with the difference that the transition from geometry to algebra in homology theory is irreversible).

In algebra, in turn, in the theory of groups (cf. extension of a group), one studies, in fact, the first and the second homology and cohomology groups. Extensive preparatory material was developed in the theory of associative algebras, the theory of Lie algebras, the theory of finite-dimensional algebras, the theory of rings and the theory of quadratic forms.

The language of homological algebra arose mainly from the process of studying homology groups. There appeared arrows as symbols for mappings and commutative diagrams (if, in a diagram, any two paths with a common beginning and end give rise to the same composite mapping, then the diagram is said to be commutative). Sequences of homomorphisms in which the kernel of each outgoing homomorphism coincides with the image of the incoming one were encountered; such sequences are called exact. It became customary to specify mathematical objects together with their mappings; the correspondences most preferred were those between objects that preserve the mappings. These correspondences became known as functors. The principal advantages of this language — the amount of information conveyed, naturalness and clarity — were soon recognized. For example, the language of homological algebra was employed ([5]) in the axiomatic exposure of the fundamentals of algebraic topology. Nowadays, this language is used in numerous studies, including those not employing homological methods.

By the mid-1940’s, homological algebra had become an independent branch of algebra. The principal domain of application of homological algebra is the category of modules over a ring. Most of the results known for modules may be applied to abelian categories with certain restrictions (this is because such categories are embeddable into categories of modules). In the most fruitful extension of the domain of application of homological algebra ([4]), the latter was extended so as to apply to arbitrary abelian categories with enough injective objects, and became applicable to arithmetical algebraic geometry and to the theory of functions in several complex variables (cf. Grothendieck category).

The principal functors of homological algebra are $ \operatorname{Hom}(A,B) $ (the group of homomorphisms of a module $ A $ to a module $ B $) and $ A \otimes B $ (the tensor product of $ A $ and $ B $). The base of the theory is the study of derived functors, which may be constructed as follows. An arbitrary module $ A $ can be represented as a quotient module of a free module $ F_{0} $, after which a similar representation $ F_{1} $ is considered for the kernel of the preceding representation, etc. One thus obtains an exact sequence: $$ \cdots \longrightarrow F_{n} \longrightarrow \cdots \longrightarrow F_{1} \longrightarrow F_{0} \longrightarrow A \longrightarrow 0. $$ An exact sequence $$ \cdots \longrightarrow P_{n} \longrightarrow \cdots \longrightarrow P_{0} \longrightarrow A \longrightarrow 0, $$ where all the modules $ P_{i} $ are projective, is called a projective resolution of the module $ A $. The application of a covariant additive functor $ T $ yields a complex, whose homology groups are known as the left derived functors of $ T $; they are denoted by $ L_{n} T $. A dual construction (for a contravariant functor) — or by using injective modules and injective resolutions (for a covariant functor) — is performed to obtain the right derived functors $ R^{n} T $. In a certain sense, the derived functors are a measure of the deviation of the functor from exactness. They are not affected by the arbitrariness involved in the construction of a resolution. To each exact sequence $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $$ correspond two infinite exact sequences of derived functors: $$ \cdots \longrightarrow {L_{n + 1} T}(C) \longrightarrow {L_{n} T}(A) \longrightarrow {L_{n} T}(B) \longrightarrow {L_{n} T}(C) \longrightarrow {L_{n - 1} T}(A) \longrightarrow \cdots $$ and $$ \cdots \longrightarrow {R^{n - 1} T}(C) \longrightarrow {R^{n} T}(A) \longrightarrow {R^{n} T}(B) \longrightarrow {R^{n} T}(C) \longrightarrow {R^{n + 1} T}(A) \longrightarrow \cdots. $$

The following notation is accepted for the derived functors of basic functors: $$ {L_{n}}(A \otimes_{R} B) = {\operatorname{Tor}^{R}_{n}}(A,B), \qquad R^{n} {\operatorname{Hom}_{R}}(A,B) = {\operatorname{Ext}_{R}^{n}}(A,B). $$ Both of these functors are functors of two arguments, $ A $ and $ B $, and for this reason, the construction of the derived functor as described above is not directly applicable to them. In such a case, one can fix one argument and construct a resolution for the other, or, having constructed resolutions of both arguments, one can construct a binary complex. The same result will be obtained in all cases. The group $ {\operatorname{Ext}_{R}^{1}}(A,B) $ is isomorphic to the group of extensions of the module $ B $ by the module $ A $ (and in this form, it has been studied for a long time). The establishment of the new relations considerably extended and advanced the theory of extensions of modules. The group $ {\operatorname{Tor}^{\mathbb{Z}}_{1}}(A,\mathbb{Q} / \mathbb{Z}) $ is the torsion part of a group $ A $. The generalization of this observation resulted in the development of the general theory of torsion.

The homology theory of algebraic systems forms part of the general scheme of derived functors. Thus, let $ \Lambda = \mathbb{Z} G $ be the group ring (cf. group algebra) of a multiplicative group $ G $ over the ring $ \mathbb{Z} $ of integers, and let $ A $ be a left and $ B $ a right $ \Lambda $-module. The study of the groups $$ {H^{n}}(G,A) = {\operatorname{Ext}_{\mathbb{Z} G}^{n}}(\mathbb{Z},A) \qquad \text{and} \qquad {H^{n}}(G,B) = {\operatorname{Tor}^{\mathbb{Z} G}_{n}}(B,\mathbb{Z}), $$ where $ \mathbb{Z} $ is considered as a trivial left $ \mathbb{Z} G $-module, is the theory of homology and cohomology of groups. Let $ L $ be a Lie algebra over a field $ \mathbb{k} $, let $ U L $ be its universal enveloping algebra, and let $ A $ be a $ U L $-module. The study of the groups $$ {H^{n}}(L,A) = {\operatorname{Ext}_{UL}^{n}}(\mathbb{k},A), $$ where $ \mathbb{k} $ is considered as a trivial $ U L $-module, is the theory of cohomology of Lie algebras. Appropriate cohomology and homology groups of monoids, abelian groups, algebras, graded algebras, rings, etc., are defined in a similar manner. The guideline in each case is the fact that the second cohomology group is the group of extensions for the type of algebraic systems under consideration.

In turn, the homology groups of algebraic systems form the subject of study of relative homological algebra.

In concrete cases, derived functors of functors are usually computed by means of an explicit resolution. The resolution may be finite (e.g., the length of the resolution of an arbitrary abelian group does not exceed $ 1 $). There has long been interest in the length of the shortest resolution (this length is called the homological dimension). The first significant result in this direction is the Hilbert syzygy theorem (appearing at the end of the 19th century). Homological-dimension theory is one of the actively-developing branches of homological algebra. The transition from modules with various restrictions of finiteness to the general case is often performed with the aid of the inductive limit and projective limit functors, $ \lim_{\rightarrow} $ and $ \lim_{\leftarrow} $. Thus, any group is the inductive limit of its finitely-generated subgroups. Every compact totally-disconnected group is representable as the projective limit of its finite quotient groups. Interest in these groups stems from their connection with Galois theory. The derived functors of these functors are used in homological dimension theory.

Derived functors for non-additive functors have been studied (e.g., functors that relate an abelian group to its group ring or to its symmetric algebra).

The principal means of computations in homological algebra, other than the resolutions already mentioned, are spectral sequences and the homology product. The former, which are a most powerful tool in the study of derived functors, approximate the homology groups of a group by the homology groups of a subgroup and a quotient group of it. The homology product relates to the study of homomorphisms of the type $$ L_{n} T \times L_{m} T \to L_{n + m} T, $$ which combine derived functors with one another.

Methods of homological algebra are now extensively employed in very different branches of mathematics, like functional analysis, the theory of functions of a complex variable, differential equations, etc. Branches of algebra such as algebraic $ K $-theory, algebraic geometry or algebraic number theory would be unthinkable without homological algebra.

References

[1]	H. Cartan, S. Eilenberg, “Homological algebra”, Princeton Univ. Press (1956). MR0077480 Zbl 0075.24305
[2]	S. MacLane, “Homology”, Springer (1963). Zbl 0818.18001 Zbl 0328.18009
[3]	H. Bass, “Algebraic $ K $-theory”, Benjamin (1968). MR249491
[4]	A. Grothendieck, “Sur quelques points d’algèbre homologique”, Tohôku Math. J., 9 (1957), pp. 119–221. MR0102537
[5]	S. Eilenberg, N.E. Steenrod, “Foundations of algebraic topology”, Princeton Univ. Press (1952). MR0050886 Zbl 0047.41402
[6]	Itogi Nauk. Ser. Mat., Algebra 1964 (1966), pp. 203–236.
[7]	N.E. Steenrood (ed.), Reviews of papers in algebraic and differential topology, topological and homological algebra, 2, Amer. Math. Soc. (1968), pp. 1174–1364.

References

[a1]	P.J. Hilton, U. Stammbach, “A course in homological algebra”, Springer (1971). MR0346025 Zbl 0238.18006