Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, say $k$. Denote by $X_y$ the closed fiber $f^{-1}(y)$ and $i:X_y \to X$ the closed immersion. We know that for any quasi-coherent sheaf $\mathcal{F}$ on $X$, there is a natural map $R^if_*\mathcal{F} \otimes k \to H^i(i^*\mathcal{F})$. The question is how does this map behave when we have the right-derived long exact sequence associated to a short exact sequence of locally free $\mathcal{O}_X$-modules? In particular, let $$0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$$ be a short exact sequence of locally free sheaves on $X$. Since the terms in the above short exact sequence are locally free it induces the following short exact sequence of locally free sheaves: $$0 \to i^*\mathcal{F}' \to i^*\mathcal{F} \to i^*\mathcal{F}'' \to 0$$ The first short exact sequence gives a long exact sequence of right derived functor, which when pulled back to $y$ gives us the following which may no longer exact: $$0 \to R^0f_*\mathcal{F}' \otimes k \to R^0f_*\mathcal{F} \otimes k \to R^0f_*\mathcal{F}'' \otimes k \to R^1f_*\mathcal{F}' \otimes k \to ...\hspace{2cm} (1)$$ And the previous short exact sequence gives the following long exact sequence of cohomology groups: $$0 \to H^0(i^*\mathcal{F}') \to H^0(i^*\mathcal{F}) \to H^0(i^*\mathcal{F}'') \to H^1(i^*\mathcal{F}') \to ...\hspace{4cm}(2)$$ We know that there is a natural map from each of the terms of the first complex to corresonding term in the second complex. The question: Is there a morphism of complexes from the first complex to the second?
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2$\begingroup$ EGA 0$_{\rm{III}}$, 12.1.1--12.1.6 (which includes the link with Cech theory). The main point is to first use topological pullback $i^{-1}$ (which is always exact!), so universal $\delta$-functor arguments are applicable, and then compose with the natural map to the ringed-space pullback $i^{\ast}$. One has a completely general "base change morphism" for higher direct images of arbitrary sheaves of modules relative to any commutative square of ringed spaces (including fiber squares of schemes as a special case). This has nothing to do with vector bundles or $\mathbf{C}$ or closed points. $\endgroup$– user27920Commented Jul 30, 2014 at 12:49
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$\begingroup$ @user52824:Thanks for the answer. Are you using that $R^if_∗\mathcal{F} \otimes k \cong H^i(\mathcal{F})$ and similarly for $\mathcal{F}′$ and $\mathcal{F}″$? $\endgroup$– user46578Commented Jul 30, 2014 at 14:32
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$\begingroup$ No, of course not (as you would realize immediately if you look at the reference I gave). It would be circular to try to prove such an isomorphism before one has already answered your question (which includes defining the base-change morphism, definitely not via Cech theory -- the latter is a means of computing what should be defined by $\delta$-functor methods). In particular, the discussion of cohomology and base change in Hartshorne's textbook on algebraic geometry is flawed in its absence of a discussion of both your question and the a-priori definition of the base change morphism. $\endgroup$– user27920Commented Jul 30, 2014 at 16:05
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$\begingroup$ @user52824: Following you first post, once we apply, $i^{-1}$, we do not have $\mathcal{O}_{X_y}$-modules. But all the results in the reference needs modules over schemes. Could you please elaborate a little bit more or your first comment. May be you could put it as an answer. It seems from your second post that this is an important point. I am sure it will be very helpful for me and may be others who have this question, since (as you point out) this is not properly covered in Hartshorne. $\endgroup$– user46578Commented Jul 30, 2014 at 21:44
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2$\begingroup$ Sure, I am well aware that $i^{-1}$ isn't valued in $O_{X_y}$-modules, but that's irrelevant: the reference does not need modules over schemes since it is applicable to arbitrary ringed spaces (and is extremely useful in that generality: complex manifolds, formal schemes, etc.). Put more effort into understanding the method in the EGA reference. It really explains everything. For example, if $f:(X',O') \rightarrow (X,O)$ is a map of ringed spaces then so is $(X',f^{-1}(O)) \rightarrow (X',O')$. Forget about schemes. I prefer not to say anything more. $\endgroup$– user27920Commented Jul 30, 2014 at 23:32
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