Grothendieck spectral sequence

If ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}$ and ${\displaystyle G\colon {\mathcal {B}}\to {\mathcal {C}}}$ are two additive and left exact functors between abelian categories such that both ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$ have enough injectives and ${\displaystyle F}$ takes injective objects to ${\displaystyle G}$-acyclic objects, then for each object ${\displaystyle A}$ of ${\displaystyle {\mathcal {A}}}$ there is a spectral sequence:

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

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

We shall use the following lemma:

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

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

which splits. This implies each ${\displaystyle B^{n+1}}$ is injective. Next we look at

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

It splits, which implies the first part of the lemma, as well as the exactness of

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

Similarly we have (using the earlier splitting):

${\displaystyle 0\to G(Z^{n})\to G(K^{n}){\overset {G(d)}{\to }}G(B^{n+1})\to 0.}$

The second part now follows. ${\displaystyle \square }$

We now construct a spectral sequence. Let ${\displaystyle A^{0}\to A^{1}\to \cdots }$ be an injective resolution of A. Writing ${\displaystyle \phi ^{p}}$ for ${\displaystyle F(A^{p})\to F(A^{p+1})}$, we have:

${\displaystyle 0\to \operatorname {ker} \phi ^{p}\to F(A^{p}){\overset {\phi ^{p}}{\to }}\operatorname {im} \phi ^{p}\to 0.}$

Take injective resolutions ${\displaystyle J^{0}\to J^{1}\to \cdots }$ and ${\displaystyle K^{0}\to K^{1}\to \cdots }$ of the first and the third nonzero terms. By the horseshoe lemma, their direct sum ${\displaystyle I^{p,\bullet }=J\oplus K}$ is an injective resolution of ${\displaystyle F(A^{p})}$. Hence, we found an injective resolution of the complex:

${\displaystyle 0\to F(A^{\bullet })\to I^{\bullet ,0}\to I^{\bullet ,1}\to \cdots .}$

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

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

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

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

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

Since ${\displaystyle H^{q}(I^{\bullet ,0})\to H^{q}(I^{\bullet ,1})\to \cdots }$ is an injective resolution of ${\displaystyle H^{q}(F(A^{\bullet }))=R^{q}F(A)}$ (it is a resolution since its cohomology is trivial),

${\displaystyle {}^{\prime }E_{2}^{p,q}=R^{p}G(R^{q}F(A)).}$

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

• 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. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.