Lyndon–Hochschild–Serre spectral sequence
Encyclopedia
In mathematics
, especially in the fields of group cohomology
, homological algebra
and number theory
the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence
relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G.
The precise statement is as follows:
Let G be a finite group
, N be a normal subgroup
. The latter ensures that the quotient G/N is a group, as well. Finally, let A be a G-module
. Then there is a spectral sequence:
The same statement holds if G is a profinite group and N is a closed normal subgroup.
The associated five-term exact sequence
is the usual inflation-restriction exact sequence
:
The spectral sequence is an instance of the more general Grothendieck spectral sequence
of the composition of two derived functors. Indeed, H∗(G, -) is the derived functor
of (−)G (i.e. taking G-invariants) and the composition of the functors (−)N and (−)G/N is exactly (−)G.
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the fields of group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
, 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...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G.
The precise statement is as follows:
Let G be a finite group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, N be a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
. The latter ensures that the quotient G/N is a group, as well. Finally, let A be a G-module
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
. Then there is a spectral sequence:
- H p(G/N, H q(N, A)) ⇒ H p+q(G, A).
The same statement holds if G is a profinite group and N is a closed normal subgroup.
The associated five-term exact sequence
Five-term exact sequence
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence.More precisely, letbe a spectral sequence, whose terms are non-trivial only for p, q ≥ 0....
is the usual inflation-restriction exact sequence
Inflation-restriction exact sequence
In mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences....
:
- 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A).
The spectral sequence is an instance of the more general Grothendieck spectral sequence
Grothendieck spectral sequence
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors G\circ F, from knowledge of the derived functors of F and G.If...
of the composition of two derived functors. Indeed, H∗(G, -) is the derived functor
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
of (−)G (i.e. taking G-invariants) and the composition of the functors (−)N and (−)G/N is exactly (−)G.
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.