Vtyjkyuk

Let $F$ be a left exact functor from the category of sheaves of abelian groups to the category of abelian groups, $mathscrF$ a sheaf of abelian groups on a topological space $X$. Since injective resolutions always exist, and acyclic ones are sent to acyclic ones, we may define the right derived functor $RF$ by $RF(mathscrF):=F(I^cdot)$ for any injective resolutions $mathscrFto I^cdot$.

However, though they exist, an injective resolution is usually messy, so we often do not use it for computations. An example is we use $checkC$ech resolution to compute for $F=(f:Xtopt)_*$; note that $R^iF(mathscrF)=H^i(X,mathscrF)$. More precisely, we pick a good open cover $mathcalU$ for $X$, and then we have $$H^i(X,mathscrF) = checkH^,i(mathcalU,mathscrF) mbox [Harshorne A.G. ex.III.4.11].$$ For good enough spaces, we can pick such a good cover and compute the right-hand side precisely.

Questions:

1. In general is there a way to compute a derived functor first by resolving by a Cech complex with a good cover?




2. If we cannot expect my first question to be true, is it at least possible for some specific functors?




3. When $F=(f:Xto Y)_*$, we have a clearer description: $R^iF(mathscrF)$ is the sheaf that associates to the presheaf
   $$Vmapsto H^i(f^^1(V), mathscrF|_f^-1(V)) mbox [Harshorne A.G. III.8.1]..$$
   Good! this makes things more explicit! I notice that this result can be obtained by my first question (if it is true). However, Hartshorne uses a quite complicated proof that refers to other concepts such as "effacable", "universal $delta$-functors". I also found that if I prove it directly by definition, it will be a mess (Homology sheaf is a quotient by the image sheaf.. so you have to take two sheafifications!). Is there a plain explanation?




4. For the third question, is it possible to get an explicit result just for $RF(mathscrF)$ but not $R^iF(mathscrF)$? I would like to know since $RF(mathscrF)$ contains more information.

1) : This is true and is a consequence of the Čech-to-derived functor spectral sequence (see the wikipedia page, where they mention that we can use this to compute derived functors).

2) : see 1)

3) I'm not sure I really understand this part, but taking a (or two) sheafification is not really a big deal, and indeed this description is really useful in particular when you are taking the stalks.

4) You can't obtain $RF(mathcal F)$ unless some really special case, since you would need an explicit resolution which is hard to produce. However there is a case when you can compute it : Deligne's highly non-trivial theorem tells you that $$Rf_* underlineBbb Q_X = bigoplus_i R^if_*underlineBbb Q_X[-i]$$
when $f : X to Y$ is a proper morphism between projective smooth varieties. Concretely this means that $Rf_* underlineBbb Q_X$ is isomorphic as a complex to its cohomology in the right degree, with zero differential. The proof uses tools from Hodge theory.

Additional remarks :

a) Let me add that projective (i.e by locally free sheaves) resolutions are more easy to find, and you can compute the full complex $RtextHom$ or the derived tensor product very explicitely. For example if $X subset Bbb P^n$ is an hypersurface, the exact sequence $$ 0 to mathcalO_Bbb P^n(-X) to mathcalO_Bbb P^n to mathcalO_X to 0$$ can be interpreted as a $2$-step projective resolution of the coherent sheaf $mathcal O_X$. Since $textHom$ is a bifunctor, you can choose to take a projective resolution, rather than an injective one :-)

b) In the derived setting, you rarely get something explicit for free, and the main (only?) tools available are spectral sequences.

c) An excellent reference is The Fourier-Mukai transfom in algebraic geometry which contains a very clear exposition of derived functors and the derived category of coherent sheaves, assuming a minimal background in homological algebra (you probably need to know what is a spectral sequence though, but if you want to understand the details you can't skip it).