There's a lemma from homological algebra that I'm using in sheaf cohomology, and I can't remember where else I've seen it. Where are some other key places it is applied?
Lemma: Let $F:\mathcal{C} \to \mathcal{D}$ is a left exact functor between abelian categories, and let $\mathcal{C}$ have enough injectives. Let $\mathscr{S}$ be a class of objects in $\mathcal{C}$ that
- Includes all injective objects
- "Completes exact sequences", i.e. if $0 \to A \to B \to C \to 0$ is exact and $A$, $B$, are in $\mathscr{S}$, then $C$ is.
- Has $R^1F(X) = 0$ for all $X$ in $\mathscr{S}$.
Then $R^iF(X)=0$ for all $X$ in $\mathscr{S}$ and all $i$.
I included the proof below, if you're interested. I've been using this to show that flabby/soft sheaves are acyclic, but I think I've seen this same cancelling long exact sequence proof before (perhaps with Ext and Tor?). Where?
For the proof, we take an object $X$ in $\mathscr{S}$, and a monomorphism from $X$ to an injective object $I$, form its cokernel $C$, and note that by hypothesis 2, the cokernel is in $\mathscr{S}$ as well. So now the long exact sequence of derived functors is $$ \require{cancel} \cancel{R^1F(A)} \to R^1F(I) \to \cancel{R^1F(C)} \to R^2F(A) \to R^2F(I) \to R^2F(C) \to \dotsb $$ All the injective objects are acyclic, so $$ \require{cancel} \cancel{R^1F(C)} \to R^2F(X) \to \cancel{R^2F(I)} \to R^2F(C) \to R^3F(A) \to \cancel{R^3F(I)} \to \dotsb, $$ which shows that $R^2F(X)=0$ for all $X$ in $\mathscr{S}$, and now we can bootstrap.
As far as I know, that's a lemma due to Grothendieck (who else?) from his Tohoku paper: See Lemma 3.3.1 in http://www.math.mcgill.ca/barr/papers/gk.pdf (english translation). It's used very often, so it's not surprising that you have seen it before - though I can't tell you where :)