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This question comes from the same place as my other one. In reading SGA 4 1/2, but not SGA4 itself (at least, not the obvious sections xv + xvi), one can learn about the "cospecialization morphisms" on cohomology. Namely, if one has a morphism $f \colon X \to B$ which is locally acyclic, and two geometric points $s, t \to B$ of which the former is a specialization of the latter, there is a map

$$\def\cosp{\operatorname{cosp}^*} \cosp \colon H^i(X_t, F|_{X_t}) \to H^i(X_s, F|_{X_s})$$

for any sheaf of torsion abelian groups $F$ on $X$. (This can be written in the six-functors language but I don't want to draw the necessary diagrams to name the maps.)

This is rather a weird map for me, in large part because no real discussion of it occurs afterwards. There is one lemma stating that if these maps are isomorphisms for all $t$, then cohomology commutes with taking stalks at $s$; and there is the theorem that this is so for a smooth, proper morphism.

The following things are unclear to me:

  • Is $\cosp$ basically (or even precisely) a nearby-cycles construction?

  • Is it compatible with base change? After all, the fibers $X_s$ and $X_t$ are, should $s$ and $t$ come from a base extension. (I tried to work the diagram but got stuck, so it seems not to be obviously so.) If not, what is the basic problem?

  • To what extent does $\cosp$ being an isomorphism implement a kind of "local to global" effect for the property of being acyclic? It appears to "spread around" the cohomology of a single fiber, as one would expect from such a thing.

  • For that matter, what is the relationship between local acyclicity and global acyclicity of $f$, aside from the obvious implication of the former by the latter?

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