In
mathematics, in the field of
homological algebra, the Grothendieck spectral sequence, introduced by
Alexander Grothendieck in his
''Tôhoku'' paper, is a
spectral sequence that computes the
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
It was noted in vari ...
s of the composition of two
functors , from knowledge of the derived functors of
and
.
Many spectral sequences in
algebraic geometry are instances of the Grothendieck spectral sequence, for example the
Leray spectral sequence.
Statement
If
and
are two additive and
left exact functors between
abelian categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
such that both
and
have
enough injectives
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categori ...
and
takes
injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categori ...
s to
-
acyclic objects, then for each object
of
there is a spectral sequence:
:
where
denotes the ''p''-th right-derived functor of
, etc., and where the arrow '
' means
convergence of spectral sequences.
Five term exact sequence
The
exact sequence of low degrees reads
:
Examples
The Leray spectral sequence
If
and
are
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, let
and
be the
category of sheaves of abelian groups on
and
, respectively.
For a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
there is the (left-exact)
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
functor
.
We also have the
global section
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
functors
:
and
Then since
and the functors
and
satisfy the hypotheses (since the direct image functor has an exact left adjoint
, pushforwards of injectives are injective and in particular
acyclic for the global section functor), the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in this case becomes:
:
for a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of abelian groups on
.
Local-to-global Ext spectral sequence
There is a spectral sequence relating the global
Ext and the sheaf Ext: let ''F'', ''G'' be
sheaves of modules
''Sheaves'' is the plural of either of two nouns:
* Sheaf (disambiguation)
* Sheave
A sheave () or pulley wheel is a grooved wheel often used for holding a belt, wire rope, or rope and incorporated into a pulley
A pulley is a wheel ...
over a
ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
; e.g., a scheme. Then
:
This is an instance of the Grothendieck spectral sequence: indeed,
:
,
and
.
Moreover,
sends injective
-modules to flasque sheaves,
which are
-acyclic. Hence, the hypothesis is satisfied.
Derivation
We shall use the following lemma:
Proof: Let
be the kernel and the image of
. We have
:
which splits. This implies each
is injective. Next we look at
:
It splits, which implies the first part of the lemma, as well as the exactness of
:
Similarly we have (using the earlier splitting):
:
The second part now follows.
We now construct a spectral sequence. Let
be an injective resolution of ''A''. Writing
for
, we have:
:
Take injective resolutions
and
of the first and the third nonzero terms. By the
horseshoe lemma, their direct sum
is an injective resolution of
. Hence, we found an injective resolution of the complex:
:
such that each row
satisfies the hypothesis of the lemma (cf. the
Cartan–Eilenberg resolution.)
Now, the double complex
gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
:
,
which is always zero unless ''q'' = 0 since
is ''G''-acyclic by hypothesis. Hence,
and
. On the other hand, by the definition and the lemma,
:
Since
is an injective resolution of
(it is a resolution since its cohomology is trivial),
:
Since
and
have the same limiting term, the proof is complete.
Notes
References
*
*
Computational Examples
* Sharpe, Eric (2003). ''Lectures on D-branes and Sheaves (pages 18–19)'',
{{PlanetMath attribution, id=1095, title=Grothendieck spectral sequence
Spectral sequences