Grothendieck spectral sequence explained

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

G\circ F , from knowledge of the derived functors of F and G .Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

Statement

If

F \colon l{A}\to l{B} and G \colon l{B}\to l{C} are two additive and left exact functors between abelian categories such that both l{A} and l{B} have enough injectives and F takes injective objects to G -acyclic objects, then for each object A of l{A} there is a spectral sequence:E ^{pq} _{2} = ({\rm R} ^{p} G \circ{\rm R} ^{q} F)(A) \Longrightarrow {\rm R} ^{p+q} (G\circ F)(A),

where

{\rm R} ^{p} G denotes the p-th right-derived functor of G , etc., and where the arrow '\Longrightarrow ' means convergence of spectral sequences.

Five term exact sequence

The exact sequence of low degrees reads

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).}

Examples

The Leray spectral sequence

See main article: Leray spectral sequence. If $X$ and $Y$ are topological spaces, let $\backslash mathcal\; =\; \backslash mathbf(X)$ and $\backslash mathcal\; =\; \backslash mathbf(Y)$ be the category of sheaves of abelian groups on $X$ and $Y$, respectively.

f \colon X \to Y there is the (left-exact) direct image functor f _{*} \colon Ab (X) \to Ab (Y) .We also have the global section functors\Gamma _{X} \colon Ab (X)\to Ab and \Gamma _{Y} \colon Ab (Y) \to Ab . Then since \Gamma _{Y} \circ f _{*} = \Gamma _{X} and the functors f _{*} and \Gamma _{Y} satisfy the hypotheses (since the direct image functor has an exact left adjoint f ^{-1} , pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:H ^{p(Y,{\rm} R} ^{q} f ^{p + q} _{* l{F})\implies H} (X, l{F}) l{F} of abelian groups on X .

Local-to-global Ext spectral sequence

(X, l{O}) ; e.g., a scheme. ThenE ^{p,q} _{2} = \operatorname{H} ^{p(X;} l{E}xt ^{q} _{l{O} }(F, G)) \Rightarrow \operatorname^_(F, G).This is an instance of the Grothendieck spectral sequence: indeed,R ^{p} \Gamma(X, - ) = \operatorname{H} ^{p(X,} - ) , R ^{q} l{H}om _{l{O} }(F, -) = \mathcalxt^q_(F, -) and R ^{n} \Gamma(X, l{H}om _{l{O} }(F, -)) = \operatorname^n_(F, -).Moreover, l{H}om _{l{O} }(F, -) sends injective l{O} -modules to flasque sheaves, which are \Gamma(X, - ) -acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma:

Proof: Let

Z ^{n,} B ^{n+1} be the kernel and the image of d: K ^{n} \to K ^{n+1} . We have0 \to Z ^{n} \to K ^{n} \overset{d}\to B ^{n+1} \to 0 , which splits. This implies each B ^{n+1} is injective. Next we look at0 \to B ^{n} \to Z ^{n} \to H ^{n(K} ^{\bullet} ) \to 0 . It splits, which implies the first part of the lemma, as well as the exactness of0 \to G(B ^{n)} \to G(Z ^{n)} \to G(H ^{n(K} ^{\bullet} )) \to 0 . Similarly we have (using the earlier splitting):0 \to G(Z ^{n)} \to G(K ^{n)} \overset{G(d)} \to G(B ^{n+1} ) \to 0 . The second part now follows. \square

We now construct a spectral sequence. Let

A ^{0} \to A ^{1} \to … be an injective resolution of A. Writing \phi ^{p} for F(A ^{p)} \to F(A ^{p+1} ) , we have:0 \to \operatorname{ker} \phi ^{p} \to F(A ^{p)} \overset{\phi ^{p}\to} \operatorname{im} \phi ^{p} \to 0 . Take injective resolutions J ^{0} \to J ^{1} \to … and K ^{0} \to K ^{1} \to … of the first and the third nonzero terms. By the horseshoe lemma, their direct sum I ^{p,} = J ⊕ K is an injective resolution of F(A ^{p)} . Hence, we found an injective resolution of the complex:0 \to F(A ^{\bullet} ) \to I ^{\bullet,} \to I ^{\bullet,} \to … . such that each row I ^{0,} \to I ^{1,} \to … satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex

E ^{p, q} _{0} = G(I ^{p,} ) gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,{} ^{\prime} E ^{p, q} _{1} = H ^{q(G(I} ^{p,} )) = R ^{q} G(F(A ^{p))} ,which is always zero unless q = 0 since F(A ^{p)} is G-acyclic by hypothesis. Hence, {} ^{\prime} E ^{n} _{2} = R ^{n} (G \circ F) (A) and {} ^{\prime} E _{2} = {} ^{\prime} E _{infty} . On the other hand, by the definition and the lemma,{} ^{\prime} E ^{p, q} _{1} = H ^{q(G(I} ^{\bullet,} )) = G(H ^{q(I} ^{\bullet,} )). Since H ^{q(I} ^{\bullet,} ) \to H ^{q(I} ^{\bullet,} ) \to … is an injective resolution of H ^{q(F(A} ^{\bullet} )) = R ^{q} F(A) (it is a resolution since its cohomology is trivial),{} ^{\prime} E ^{p, q} _{2} = R ^{p} G(R ^{qF(A)).} Since {} ^{\prime} E _{r} and {} ^{\prime} E _{r} have the same limiting term, the proof is complete. \square

References

Computational Examples

• Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19),