Last semester, I taught a course *Sheaves in topology*, about the cohomology of topological spaces via sheaves. An important and often-used tool is the *Grothendieck spectral sequence*: if

are left exact functors of abelian categories, with composition , where and have enough injectives, and takes injectives to -acyclic objects, then there is a spectral sequence

for any .

However, I do not want to assume spectral sequences, nor spend multiple lectures developing them from scratch. Here is a variant that is easy to prove and often sufficient:

**Lemma.** *Let and as above, and let .*

*If for all and all , then*

*for all .**If for all and all , then*

*for all .*

*Proof.* For both parts, let be an exact sequence with injective. The long exact sequences for and give

and isomorphisms and for . Write for the image of , so that we get short exact sequences

(1)

Since is -acyclic, we get

and isomorphisms for .

We prove (1) by induction on . For , there is nothing to prove. Since , , and are -acyclic, so are and by the short exact sequences of (1), and we get exact sequences

These glue to an exact sequence , which by the exact sequence for gives . This proves the result for . We also see that is acyclic for all : for , we proved this above, and for this follows from . Thus satisfies the same hypotheses as , and the isomorphisms

for prove the result by induction on .

Likewise, we prove (2) by induction on . Again, for , there is nothing to prove. The long exact sequence of gives for all since for all . We obtain an exact sequence

and isomorphisms for . Comparing with the long exact sequence for gives , showing the result for . Moreover, we have for all and all , so satisfies the same hypotheses as . Then the isomorphisms

show the result for by induction.

**Corollary.** *If is exact and takes injectives to -acyclic objects, then*

*for all . Likewise, if is exact, then the acyclicity assumption on is automatic, and*

*for all . *

**Example.** This is used many times in sheaf cohomology. Some examples:

- If is a closed subset, then for all abelian sheaves on and all .
- If is a locally closed subset, then for all abelian sheaves on .
- If is the derived functor of the inclusion from sheaves into presheaves, then .
- If is a sheaf on , then the sheafification of is for .
- The higher pushfoward along a map is the sheafification of .
- If is a map and for , then . This is used for the projection (with a constant sheaf) in the proof of homotopy invariance of .
- If is a sheaf and is an open cover such that for all and all finite subsets , then for all . (Here, we write for .)
- If is a topological space such that for implies for all , then for all . (It’s not hard to show that this hypothesis is satisfied if is paracompact Hausdorff.)

It’s a good exercise to go through all of these and see how they follow from the corollary.