Slogan: Smooth base change for torsion abelian etale sheaf is what flat base change for quasi-cohrent sheaf.
Idea: Let is a morphism of a proper complex analytic variety into the disk. In practice, for small enough, is usually a locally trivial fibration (but maybe not over the entire closed interval , c.f. degeneration of curves in smooth families). Then is a homotopy equivalence. If satisfies and for , then using the Leray spectral sequence for we can show is an isomorphism. As a result, we can define a cospecialization map
To calculate at a point (the only interesting place), we can take a small ball centered at of radius ; and for small enough, consider the homology cycle . This is the vanishing cycle at (think of a loop around a smooth hyperboloid that degenerates into a double cone). Then we have Thus the cospecialization morphism is defined as long as and for . We say that is locally acyclic.
If is a scheme and is a geometric point. Let be a geometric point, we say is the generisation of and is the specialization of .
If is a morphism of schemes and is a geometric point of , then is a geometric point. Consider the base change , we call it the variety of vanishing cycles. We call is locally acyclic at if for every , the reduced cohomology of the constant sheaf on vanishes for every invertible in , i.e. and for , and is locally acyclic if it is locally acyclic at every .
Lemma 1 (Stack project 0GJS): Local acyclicity is closed under quasi-finite base change. More generally, it is closed under base change along where is an inverse limit of quasi-finite -schemes with affine transition morphisms .
The idea (and the proof provided by Deligne) is that every vanishing cycle of is a vanishing cycle of .
Lemma 2 (SGA 4.5, V-3, Lemma 1.5, a bit terse; for the full detail, see section 2.9 of Aaron Landesman’s note instead) In the following Cartesian diagram, we have and the higher pushforward for .
Again the proof given by Deligne is a bit terse, and you probably don’t understand why he considers using the normalization. From my understanding the importance stems from the fact that the etale local ring of a normal scheme is a domain (by applying this and this).
Using Lemma 2, we can define a cospecialization map as follows: Consider the Cartesian diagram. Note that is locally acyclic by Lemma 1 (note that we need the more general version). We can define by
The first arrow (isomorphism) is due to Lemma 2 (and Leray spectral sequence), which tells us that for . The second one is because the restriction of to is (by local acyclicity and just put .)
Remark: If is the normalization of in , and then we have , which shows the utility of normalization. The importance of normality is manifested in Zariski’s main theorem, see the excellent explanation of its underlying geometric content here.
Theorem 3 Suppose is a locally Noetherian scheme, a geometric point of and a morphism. We suppose
The morphism is locally acyclic.
For every geometric point of and every , the cospecialization map is bijective.
Then the canonical homomorphism is bijective for every .
The proof uses the following reduction: First the question is a local one so we can suppose . Then it suffices to show that for every sheaf of -modules on , the homomophism is bijective.
Every sheaf of -modules is filtered inductive limit of constructible sheaves of -modules (stack project, 0F0N). Moreover, every constructible sheaf embeds into a sheaf of the form where is a finite collection of generisations of (i.e. geometric point of ) and is a finite free -module on (Stack project, 09Z6). Note that Lemma 2 and the condition (b) implies that for , the homomorphism is bijective (For , we have and for both sides vanish. Note that condition (b) is crucial, e.g. is smooth hence locally acyclic but at LHS is one-dimensional while RHS is zero-dimensional.) The following purely homological algebra lemma finishes the job:
Lemma 4 If is an abelian category in which filtered inductive limit exists, is a map of -functors that vanishes in degrees from to commuting with filtered colimits. Suppose there exists two subsets and of objects of such that
every object of is an filtered colimit of objects in ,
every object belonging to is a subobject of an object belonging to .
Then TFAE:
is bijective for every and for every .
is bijective for every and every .
The proof is by induction on and a repeated use of five-lemma.
We deduce two corollaries from this theorem:
Corollary 5: If , and is a locally acyclic morphism. Suppose for every geometric point of the fiber is acyclic (i.e. ). Then we have and for .
Corollary 6: Composite of locally acyclic morphisms are locally acyclic. More precisely, if and are morphisms of locally noetherian schemes. If and are locally acyclic, so is .
To see that corollary 6 follows from corollary 5, we can suppose and are strictly local and and are local morphisms. We need to show if is a geometric point of , then we have . Since is locally acyclic, we have . In addition the morphism are locally acyclic and the geometric fibers are ayclic because is locally acyclic. Then by corollary 5, we have for and are the constant sheaf on . We can now conclude using Leray’s spectral sequence.
Theorem 7 Smooth morphisms are locally acyclic.
Since the assertion is local for the etale topology on and , we can suppose . By passage to the limit, we can suppose is Noetherian and transitivity of local acyclicity allows us to further reduce to the case that . Renaming and , where is the henselization of at , our task is to show the geometric fibers of are acyclic.
If is a geometric point of , the fiber is projective limit of affine smooth curves on , and so for by the theory of cohomology of curves. Thus we only need to show and .
To show , it reduces to the following proposition in commutative algebra:
Proposition 8 If is a strictly local Henselian ring, and , then the geomtric fibers of are connected.
By passage to the limit we can reduce to the case that is a strictly Henselization of a finitely generated -algebra. The importance of this reduction is that will then be an excellent ring. What do we gain? Note that it suffices to show for every where is a finite separable subextension of , the fiber is connected. For this we want to reduce to normal, becuse then will be normal and every localization at a prime will then be a normal domain, so its spectrum is integral, in particular connected. To reduce to this case it essentially boils down to verifying is an isomorphism where is the normalization of in , and since is excellent, will be finite over . The claim then follows by observing the ring on the left is Henselian local and filtered colimit of etale local algebras on . (For details see Lemma 2.6.2 of Aaron Landesman’s note).
To show , it suffices to prove the following which is a restatement using the torsor interpretation of .
Proposition 9 If is a strictly local Henselian ring, and , and a geometric point of . Then every -torsor over in the etale topology is trivial if in the residue field of .
The assumption that in the residue field of is necessary, c.f. the Artin-Schreier cover is a nontrivial connected -torsor when is a separably closed field of characteristic .
http://math.stanford.edu/~conrad/papers/nagatafinal.pdf
Pushforward along finite morphism is exact: https://stacks.math.columbia.edu/tag/03QP
Relative normalization: https://stacks.math.columbia.edu/tag/0BAK