In homological algebra

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 and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence

0 → P → Q → R → 0

of abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, when they are introduced as coefficients into a chain complex

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

 C, and which appears in the homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 groups as a homomorphism reducing degree by one,

β: H_i(C, R) → H_{i − 1}(C, P).

To be more precise, C should be a complex of free

Free abelian group

In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

 with C (some flat module

Flat module

In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...

 condition should enter). The construction of β is by the usual argument (snake lemma

Zig-zag lemma

In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes...

).

A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have

β: Hⁱ(C, R) → H^{i + 1}(C, P).

The Bockstein homomorphism β of the coefficient sequence

0 → Z/pZ → Z/p²Z → Z/pZ → 0

is used as one of the generators of the Steenrod algebra

Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...

. This Bockstein homomorphism has the two properties

ββ = 0 if p>2

β(a∪b) = β(a)∪b + (-1)^{dim a} a∪β(b)

in other words it is a superderivation acting on the cohomology mod p of a space.