Английская Википедия:Grothendieck spectral sequence
Шаблон:Short description 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 functors of the composition of two functors <math> G\circ F</math>, from knowledge of the derived functors of <math>F</math> and <math>G</math>. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
Statement
If <math>F \colon\mathcal{A}\to\mathcal{B}</math> and <math>G \colon \mathcal{B}\to\mathcal{C}</math> are two additive and left exact functors between abelian categories such that both <math>\mathcal{A}</math> and <math>\mathcal{B}</math> have enough injectives and <math>F</math> takes injective objects to <math>G</math>-acyclic objects, then for each object <math>A</math> of <math>\mathcal{A}</math> there is a spectral sequence:
- <math>E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A),</math>
where <math>{\rm R}^p G</math> denotes the p-th right-derived functor of <math>G</math>, etc., and where the arrow '<math>\Longrightarrow</math>' means convergence of spectral sequences.
Five term exact sequence
The exact sequence of low degrees reads
- <math>0\to {\rm R}^1G(FA)\to {\rm R}^1(GF)(A) \to G({\rm R}^1F(A)) \to {\rm R}^2G(FA) \to {\rm R}^2(GF)(A).</math>
Examples
The Leray spectral sequence
Шаблон:Main If <math display="inline">X</math> and <math display="inline">Y</math> are topological spaces, let <math display="inline">\mathcal{A} = \mathbf{Ab}(X)</math> and <math display="inline">\mathcal{B} = \mathbf{Ab}(Y)</math> be the category of sheaves of abelian groups on <math display="inline">X</math> and <math display="inline">Y</math>, respectively.
For a continuous map <math>f \colon X \to Y</math> there is the (left-exact) direct image functor <math>f_* \colon \mathbf{Ab}(X) \to \mathbf{Ab}(Y)</math>. We also have the global section functors
- <math>\Gamma_X \colon \mathbf{Ab}(X)\to \mathbf{Ab}</math> and <math>\Gamma_Y \colon \mathbf{Ab}(Y) \to \mathbf {Ab}.</math>
Then since <math>\Gamma_Y \circ f_* = \Gamma_X</math> and the functors <math> f_*</math> and <math>\Gamma_Y</math> satisfy the hypotheses (since the direct image functor has an exact left adjoint <math>f^{-1}</math>, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:
- <math>H^p(Y,{\rm R}^q f_*\mathcal{F})\implies H^{p+q}(X,\mathcal{F})</math>
for a sheaf <math>\mathcal{F}</math> of abelian groups on <math>X</math>.
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 over a ringed space <math>(X, \mathcal{O})</math>; e.g., a scheme. Then
- <math>E^{p,q}_2 = \operatorname{H}^p(X; \mathcal{E}xt^q_{\mathcal{O}}(F, G)) \Rightarrow \operatorname{Ext}^{p+q}_{\mathcal{O}}(F, G).</math>[1]
This is an instance of the Grothendieck spectral sequence: indeed,
- <math>R^p \Gamma(X, -) = \operatorname{H}^p(X, -)</math>, <math>R^q \mathcal{H}om_{\mathcal{O}}(F, -) = \mathcal{E}xt^q_{\mathcal{O}}(F, -)</math> and <math>R^n \Gamma(X, \mathcal{H}om_{\mathcal{O}}(F, -)) = \operatorname{Ext}^n_{\mathcal{O}}(F, -)</math>.
Moreover, <math>\mathcal{H}om_{\mathcal{O}}(F, -)</math> sends injective <math>\mathcal{O}</math>-modules to flasque sheaves,[2] which are <math>\Gamma(X, -)</math>-acyclic. Hence, the hypothesis is satisfied.
Derivation
We shall use the following lemma:
Proof: Let <math>Z^n, B^{n+1}</math> be the kernel and the image of <math>d: K^n \to K^{n+1}</math>. We have
- <math>0 \to Z^n \to K^n \overset{d}\to B^{n+1} \to 0,</math>
which splits. This implies each <math>B^{n+1}</math> is injective. Next we look at
- <math>0 \to B^n \to Z^n \to H^n(K^{\bullet}) \to 0.</math>
It splits, which implies the first part of the lemma, as well as the exactness of
- <math>0 \to G(B^n) \to G(Z^n) \to G(H^n(K^{\bullet})) \to 0.</math>
Similarly we have (using the earlier splitting):
- <math>0 \to G(Z^n) \to G(K^n) \overset{G(d)} \to G(B^{n+1}) \to 0.</math>
The second part now follows. <math>\square</math>
We now construct a spectral sequence. Let <math>A^0 \to A^1 \to \cdots</math> be an injective resolution of A. Writing <math>\phi^p</math> for <math>F(A^p) \to F(A^{p+1})</math>, we have:
- <math>0 \to \operatorname{ker} \phi^p \to F(A^p) \overset{\phi^p}\to \operatorname{im} \phi^p \to 0.</math>
Take injective resolutions <math>J^0 \to J^1 \to \cdots</math> and <math>K^0 \to K^1 \to \cdots</math> of the first and the third nonzero terms. By the horseshoe lemma, their direct sum <math>I^{p, \bullet} = J \oplus K</math> is an injective resolution of <math>F(A^p)</math>. Hence, we found an injective resolution of the complex:
- <math>0 \to F(A^{\bullet}) \to I^{\bullet, 0} \to I^{\bullet, 1} \to \cdots.</math>
such that each row <math>I^{0, q} \to I^{1, q} \to \cdots</math> satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)
Now, the double complex <math>E_0^{p, q} = G(I^{p, q})</math> gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
- <math>{}^{\prime \prime} E_1^{p, q} = H^q(G(I^{p, \bullet})) = R^q G(F(A^p))</math>,
which is always zero unless q = 0 since <math>F(A^p)</math> is G-acyclic by hypothesis. Hence, <math>{}^{\prime \prime} E_{2}^n = R^n (G \circ F) (A)</math> and <math>{}^{\prime \prime} E_2 = {}^{\prime \prime} E_{\infty}</math>. On the other hand, by the definition and the lemma,
- <math>{}^{\prime} E^{p, q}_1 = H^q(G(I^{\bullet, p})) = G(H^q(I^{\bullet, p})).</math>
Since <math>H^q(I^{\bullet, 0}) \to H^q(I^{\bullet, 1}) \to \cdots</math> is an injective resolution of <math>H^q(F(A^{\bullet})) = R^q F(A)</math> (it is a resolution since its cohomology is trivial),
- <math>{}^{\prime} E^{p, q}_2 = R^p G(R^qF(A)).</math>
Since <math>{}^{\prime} E_r</math> and <math>{}^{\prime \prime} E_r</math> have the same limiting term, the proof is complete. <math>\square</math>
Notes
References
Computational Examples
- Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19), Шаблон:Arxiv
