Leray spectral sequence
Encyclopedia
In mathematics
, the Leray spectral sequence was a pioneering example in homological algebra
, introduced in 1946 by Jean Leray
. The formulation was of a spectral sequence
, expressing the relationship holding in sheaf cohomology
between two topological space
s X and Y, and set up by a continuous mapping
At the time of Leray's work, neither of the two concepts involved (spectral sequence, sheaf cohomology) had reached anything like a definitive state. Therefore it is rarely the case that Leray's result is quoted in its original form. After much work in the seminar of Henri Cartan
, in particular, a statement was reached of this kind: assuming some hypotheses on X and Y, and a sheaf F on X, there is a direct image sheaf
on Y.
There are also higher direct images
The E2 term of the typical Leray spectral sequence is
The required statement is that this abuts to the sheaf cohomology
In the formulation achieved by Alexander Grothendieck
by about 1957, this is the Grothendieck spectral sequence
for the composition of two derived functor
s.
Earlier (1948/9) the implications for singular cohomology were extracted as the Serre spectral sequence
, which makes no use of sheaves.
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...
, the Leray spectral sequence was a pioneering example 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...
, introduced in 1946 by Jean Leray
Jean Leray
Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
. The formulation was of 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...
, expressing the relationship holding in sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
between two topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s X and Y, and set up by a continuous mapping
- f:X → Y.
At the time of Leray's work, neither of the two concepts involved (spectral sequence, sheaf cohomology) had reached anything like a definitive state. Therefore it is rarely the case that Leray's result is quoted in its original form. After much work in the seminar of Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
, in particular, a statement was reached of this kind: assuming some hypotheses on X and Y, and a sheaf F on X, there is a direct image sheaf
- f∗F
on Y.
There are also higher direct images
- Rqf∗F.
The E2 term of the typical Leray spectral sequence is
- Hp(Y, Rqf∗F).
The required statement is that this abuts to the sheaf cohomology
- Hr(X, F).
In the formulation achieved by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
by about 1957, this is the 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...
for the composition of two 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 :...
s.
Earlier (1948/9) the implications for singular cohomology were extracted as the Serre spectral sequence
Serre spectral sequence
In mathematics, the Serre spectral sequence is an important tool in algebraic topology...
, which makes no use of sheaves.