Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. The hidden fabric of mathematics is woven of chain complexes, which manifest themselves through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.

From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does noncommutative geometry of Alain Connes.

Contents

Chain complexes and homology

Main article: Chain complex

The chain complex is the central notion of homological algebra. It is a sequence (C_\bullet, d_\bullet) of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero:

C_\bullet: \cdots \to C_{n+1} \begin{matrix} d_{n+1} \\ \longrightarrow \\ \, \end{matrix} C_n \begin{matrix} d_n \\ \longrightarrow \\ \, \end{matrix} C_{n-1} \begin{matrix} d_{n-1} \\ \longrightarrow \\ \, \end{matrix} \cdots, \quad d_n \circ d_{n+1}=0.

The elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials. The chain groups Cn may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of R-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category Ab of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to any abelian category. Every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

B_n \subseteq Z_n \subseteq C_n.

Subgroups of abelian groups are automatically normal; therefore we can define the nth homology group Hn(C) as the factor group of the n-cycles by the n-boundaries,

H_n(C) = Z_n/B_n = \operatorname{Ker}\, d_n/ \operatorname{Im}\, d_{n+1}.

A chain complex is called acyclic or an exact sequence if all its homology groups are zero.

Chain complexes arise in abundance in algebra and algebraic topology. For example, if X is a topological space then the singular chains Cn(X) are formal linear combinations of continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices of X; if A = F/R is a presentation of an abelian group A by generators and relations, where F is a free abelian group spanned by the generators and R is the subgroup of relations, then letting C1(A) = R, C0(A) = F, and Cn(A) = 0 for all other n defines a sequence of abelian groups. In all these cases, there are natural differentials dn making Cn into a chain complex, whose homology reflects the structure of the topological space X, the simplicial complex K, or the abelian group A. In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds.

On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, R-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.

Functoriality

A continuous map of topological spaces gives rise to a homomorphism between their nth homology groups for all n. This basic fact of algebraic topology finds a natural explanation through certain properties of chain complexes. Since it is very common to study several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.

A morphism between two chain complexes, F: C_\bullet\to D_\bullet, is a family of homomorphisms of abelian groups Fn:Cn → Dn that commute with the differentials, in the sense that Fn -1 •  dnC = dnD • Fn for all n. A morphism of chain complexes induces a morphism H_\bullet(F) of their homology groups, consisting of the homomorphisms Hn(F): Hn(C) → Hn(D) for all n. A morphism F is called a quasi-isomorphism if it induces the identity map on the nth homology for all n.

Many constructions of chain complexes arising in algebra and geometry, including singular homology, have the following functoriality property: if two objects X and Y are connected by a map f, then the associated chain complexes are connected by a morphism F = C(f) from C_\bullet(X) to C_\bullet(Y), and moreover, the composition g • f of maps f: X → Y and g: Y → Z induces the morphism C(g • f) from C_\bullet(X) to C_\bullet(Z) that coincides with the composition C(g) • C(f). It follows that the homology groups H_\bullet(C) are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes L_\bullet, M_\bullet, N_\bullet and two morphisms between them, f:L_\bullet\to M_\bullet, g: M_\bullet\to N_\bullet, is called an exact triple, or a short exact sequence of complexes, and written as

0\rightarrow L_\bullet \begin{matrix} f \\ \longrightarrow \\ \, \end{matrix} M_\bullet \begin{matrix} g \\ \longrightarrow \\ \, \end{matrix} N_\bullet \rightarrow 0,

if for any n, the sequence

0\rightarrow L_n \begin{matrix} f_n \\ \longrightarrow \\ \, \end{matrix} M_n \begin{matrix} g_n \\ \longrightarrow \\ \, \end{matrix} N_n \rightarrow 0

is a short exact sequence of abelian groups. By definition, this means that fn is an injection, gn is a surjection, and Im fn =  Ker gn. One of the most basic theorems of homological algebra states that, in this case, there is a long exact sequence in homology,

\ldots\rightarrow H_n(L) \begin{matrix} H_n(f) \\ \longrightarrow \\ \, \end{matrix} H_n(M) \begin{matrix} H_n(g) \\ \longrightarrow \\ \, \end{matrix} H_n(N)\, \begin{matrix} \delta_n \\ \rightarrow \\ \, \end{matrix} \, H_{n-1}(L) \begin{matrix} H_{n-1}(f) \\ \longrightarrow \\ \, \end{matrix} H_{n-1}(M) \rightarrow\ldots,

where the homology groups of L, M, and N cyclically follow each other, and δn are certain homomorphisms determined by f and g, called the connecting homomorphisms. Topological manifestations of this theorem include the Mayer-Vietoris sequence and the long exact sequence for relative homology.

Foundational aspects

Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C*-algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology.

Central to homological algebra is the notion of exact sequence; these can be used to perform actual calculations. A classical tool of homological algebra is that of derived functor; the most basic examples are functors Ext and Tor.

With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows:

These move from computability to generality.

The computational sledgehammer par excellence is the spectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.

There have been attempts at 'non-commutative' theories which extend first cohomology as torsors (important in Galois cohomology).

References