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After reading Section III.11 (The theorem on Formal functions) and III.12 (The semicontinuity Theorem), I feel that I get some kind of formalism instead of a clear picture of what is going on. So the big question is how one should understand these two sections. What is the morale behind all the technical endeavor? I especially want to compare this with the section "proper base change theorem" in Milne's online note on étale cohomology (I.17).

In the étale case, if $X\to Y$ is proper, $\mathcal{F}$ is a torsion sheaf on the site X_{ét}, and the following diagram is Cartesian, then $u^*R^if_* \mathcal{F}\cong R^ig_*(v^* \mathcal{F})$. \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ll} X' & \ra{v} & X \\ \da{g} & & \da{f} \\ Y' & \ra{u} & Y \\ \end{array}

But in Hartshorne's case, we are dealing with Zariski topology so things are not so nice. For example, it seems to me that Theorem III.11.1 in Hartshornes says that the base change is "true" if we look at infinitesimal neighborhoods of a point in $Y$. I hope someone could make some more remarks (Taking completion is like passing to the infinitesimal neighborhoods, that's pretty much all I understand here. )

For all that follows, let's assume that $Y=\mathrm{Spec}(A)$, and Y'=\mathrm{Spec}(B), and $\mathcal{F}$ a quasicoherent sheaf on $X$. In section III.12, Hartshorne defined a functor $T^i(M)=H^i(X, \mathcal{F}\otimes_A M)$ for any $A$-module $M$. I don't think I quite understand this functor. I see that $T^i(A)\otimes B$ is the module associated with the sheaf $u^*R^if_*\mathcal{F}$. If $B=k(y)$, Corollary 9.4 tells me that $T^i(k(y))$ is the module for $R^i g_*(v^*\mathcal{F})$. But for a general A-algebra $B$, is there any meaning to $T^i(B)$? It seems to be a bit odd that a lot of the work is done for a general $M$, then in the end, all the major theorems restrict to the case $M=k(y)$. For example, in Theorem 12.11, can one go beyond just the fibers and write down some statement like the étale case (with more conditions, certainly)?

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    @Jianwei: Sorry, I'm actually wrong. Even for quasi-coherent sheaves, this is apparently false because the scheme $X'$ itself should be taken in the "derived" sense! If one works with derived schemes, then the base-change isomorphism is an isomorphism (even without properness hypotheses). (I just checked this claim with Dennis Gaitsgory and he says there is no source at this point...)2011-06-24

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