Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in 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.

If $F:{\mathcal {A}}\to {\mathcal {B}}$ and $G:{\mathcal {B}}\to {\mathcal {C}}$ are two additive and left exact functors between abelian categories such that $F$ takes F-acyclic objects (e.g., injective objects) to $G$ -acyclic objects and if ${\mathcal {B}}$ has enough injectives, then there is a spectral sequence for each object $A$ of ${\mathcal {A}}$ that admits an F-acyclic resolution:

E_{2}^{{pq}}=({{\rm {R}}}^{p}G\circ {{\rm {R}}}^{q}F)(A)\Longrightarrow {{\rm {R}}}^{{p+q}}(G\circ F)(A).

Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

The exact sequence of low degrees reads

0 → R¹G(FA) → R¹(GF)(A) → G(R¹F(A)) → R²G(FA) → R²(GF)(A).

Examples

The Leray spectral sequence

If $X$ and $Y$ are topological spaces, let

{\mathcal {A}}={\mathbf {Ab}}(X)

and

{\mathcal {B}}={\mathbf {Ab}}(Y)

be the category of sheaves of abelian groups on X and Y, respectively and

{\mathcal {C}}={\mathbf {Ab}}

be the category of abelian groups.

For a continuous map

f:X\to Y

there is the (left-exact) direct image functor

f_{*}:{\mathbf {Ab}}(X)\to {\mathbf {Ab}}(Y)

We also have the global section functors

\Gamma _{X}:{\mathbf {Ab}}(X)\to {\mathbf {Ab}}

and

\Gamma _{Y}:{\mathbf {Ab}}(Y)\to {\mathbf {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_{*}{\mathcal {F}})\implies H^{{p+q}}(X,{\mathcal {F}})

for a sheaf ${\mathcal {F}}$ of abelian groups on $X$ , and this is exactly the Leray spectral sequence.

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 $(X,{\mathcal {O}})$ ; e.g., a scheme. Then

E_{2}^{{p,q}}=\operatorname {H}^{p}(X;{\mathcal {E}}xt_{{{\mathcal {O}}}}^{q}(F,G))\Rightarrow \operatorname {Ext}_{{{\mathcal {O}}}}^{{p+q}}(F,G).

^[1]

This is an instance of the Grothendieck spectral sequence: indeed,

R^{p}\Gamma (X,-)=\operatorname {H}^{p}(X,-)

R^{q}{\mathcal {H}}om_{{{\mathcal {O}}}}(F,-)={\mathcal {E}}xt_{{{\mathcal {O}}}}^{q}(F,-)

and

R^{n}\Gamma (X,{\mathcal {H}}om_{{{\mathcal {O}}}}(F,-))=\operatorname {Ext}_{{{\mathcal {O}}}}^{n}(F,-)

Moreover, ${\mathcal {H}}om_{{{\mathcal {O}}}}(F,-)$ sends injective ${\mathcal {O}}$ -modules to flasque sheaves,^[2] which are $\Gamma (X,-)$ -acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma:

Lemma — If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n,

H^{n}(K^{{\bullet }})

is an injective object and for any left-exact additive functor G on C,

H^{n}(G(K^{{\bullet }}))=G(H^{n}(K^{{\bullet }})).

Proof: Let $Z^{n},B^{{n+1}}$ be the kernel and the image of $d:K^{n}\to K^{{n+1}}$ . We have

0\to Z^{n}\to K^{n}{\overset {d}\to }B^{{n+1}}\to 0

which splits and implies $B^{{n+1}}$ is injective and the first part of the lemma. Next we look at

0\to B^{n}\to Z^{n}\to H^{n}(K^{{\bullet }})\to 0.

It splits. Thus,

0\to G(B^{n})\to G(Z^{n})\to G(H^{n}(K^{{\bullet }}))\to 0.

Similarly we have (using the early 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 $A^{0}\to A^{1}\to \cdots$ be an F-acyclic 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 $J^{0}\to J^{1}\to \cdots$ and $K^{0}\to K^{1}\to \cdots$ of the first and the third nonzero terms. By the horseshoe lemma, their direct sum $I^{{p,\bullet }}=J\oplus K$ 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 ,0}}\to I^{{\bullet ,1}}\to \cdots .

such that each row $I^{{0,q}}\to I^{{1,q}}\to \cdots$ satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex $E_{0}^{{p,q}}=G(I^{{p,\bullet }})$ gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,

{}^{{\prime \prime }}E_{1}^{{p,q}}=H^{q}(G(I^{{p,\bullet }}))=R^{q}G(F(A^{p}))

which is always zero unless q = 0 since $F(A^{p})$ is G-acyclic by hypothesis. Hence, ${}^{{\prime \prime }}E_{{2}}^{n}=R^{n}(G\circ F)(A)$ and ${}^{{\prime \prime }}E_{2}={}^{{\prime \prime }}E_{{\infty }}$ . On the other hand, by the definition and the lemma,

{}^{{\prime }}E_{1}^{{p,q}}=H^{q}(G(I^{{\bullet ,p}}))=G(H^{q}(I^{{\bullet ,p}})).

Since $H^{q}(I^{{\bullet ,0}})\to H^{q}(I^{{\bullet ,1}})\to \cdots$ 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_{2}^{{p,q}}=R^{p}G(R^{q}F(A)).

Since ${}^{{\prime }}E_{r}$ and ${}^{{\prime \prime }}E_{r}$ have the same limiting term, the proof is complete. $\square$

Notes

↑ Godement, Ch. II, Theorem 7.3.3.
↑ Godement, Ch. II, Lemma 7.3.2.

References

Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259. MR 1269324.

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