This is a continuation of this post.
Cohomology with supports
We now wish to move on from the proper case of Proposition 2 to the case of non-proper smooth connected curves . Now, Lemma 1 (and the fact that Proposition 2 is stated in terms of it) strongly suggest that we should try and compute the cohohomology of by embedding it into a proper smooth connected curve and somehow understand what the ‘difference’ between the cohomology of and is.
Namely, we want a way to explicitly compare the cohomology of a scheme and an open subset . Our intuition strongly pushes us to imagine that somehow their should be a relationship involving the cohomology for , , and (the closed complement of ). It turns out that things are slightly more complicated than one might initially imagine though.
So, before we explain the precise way in which we compare these three cohomologies, let’s discuss a model sucess in a slightly modified situation. Namely, instead of working with cohomology itself let us instead think of compactly supported cohomology. This should be a reasonable thing to do since (assuming that we’re interested in smooth varieties) Poincare duality (e.g. see [Mil, Corollary 11.2]) says that compactly supported cohomology is essentially dual to normal cohomology.
Now, while compactly supported cohomology has a lot of lovably annoying functorial indiosyncracies (it’s not functorial for arbitrary maps and when it is it can be covariant or contravariant depending on the property of the map) one thing that compactly supported cohomology was built to do is understand our question of how the cohomology of , , and are related. Namely, we have the following simple theorem:
Observation 21: Let be an algebraically closed field and let be proper variety over , and open subscheme, and its complement. Then, for each prime and each there is a long exact sequence in compactly supported cohomology
Proof: For example one can see [Mil, Remark III.1.30].
Remark 22: Note that as exemplified in Observation 21, and as will persist below, one can generally be pretty loose with what scheme structure one is taking on a closed subscheme. Namely, if is a closed subset of a scheme then essentially all the scheme structures on have the same underlying reduced scheme. But, the passage from to doesn’t change the etale topology by the topological invariance of the etale site (e.g. see [Stacks, Tag03SI]).
It is a result like Observation 21 that we’d like to have for normal cohomology. Note that it will necessarily need to be more complicated since the weird functorial properties of compactly supported cohomology made the above possible: namely we have maps from and both in to but maps on cohomology in different directions. This phenomenon cannot occur for normal cohomology which is always contravariantly functorial. So, the open embedding gives rise to a map
and while it wouldn’t make sense for the kernel of this map to be it could, hopefully, still be some cohomology group related to .
In particular, we already know what the answer should be at least for . Namely, for an arbitrary sheaf we can certainly set
In words one should think of as the ‘sections of with supports in ‘. One can check that the functor
is left exact and we define to be its -derived functor. We call this the cohomology with supports in .
As promised, and essentially as constructed, we have the following:
Proposition 23: Let be a scheme, an open subscheme, and its closed complement. Then, for any abelian sheaf on there is a long exact sequence of abelian groups
Proof: This is [Mil, Proposition III.1.25] or [Fu, Proposition 5.6.11].
The last thing we mention in this subsection is that there is a natural way to interpret the group in terms of the theory of torsors (e.g. see the notes from this post). Namely, one can show that
Here two pairs and are isomorphic if there is an isomorphism so that . With this interpretation the map
can be interpreted as sending to the pair where is the multiplication by map. Similarly, the map
can be thought of as the map which associates to a pair the torsor .
Relation to shriek backs and a spectral sequence
Even though it’s not strictly necessary for our computations, I think it’s useful to explain a somewhat confusing point (or at least it was confusing to me the first time I learned this stuff) about cohomology with supports. Namely, we explicitly built as follows:
one then might wonder what the relationship between cohomology with supports and the sheaf .
In particular, an incredibly naive guess one might have is that is the cohomology of , but this is totally wrong. For example, note that, by definition we have an exact sequence
and so we get a long exact sequence that looks like
and if we want this to match Proposition 23 we’d want to match but this is horribly, horribly wrong. For example, in the case where is an affine open subcurve of the proper connected smooth curve then one has that and so
which, as Proposition 2 tells us, is quite different from in general.
That said, there is a relationship between cohomology, , and . Namely, let us set and call it the shriek back of along . This scary terminology comes from the fact that is an adjoint pair (e.g. see [Fu, Proposition 5.4.2]) where, in general, for a map we define so that is an adjoint pair. One can show that is left exact (e.g. see loc. cit.) and thus we have a well-defined notion of the right-defined functors
which are often, somewhat confusingly, written as . In particular, note that is a sheaf on .
Now, as one can check one has that and so, in particular, we see that
so we see that the composition
is precisely . In particular since (as one can check) preserves injectives we obtain the following from the Grothendieck spectral sequence:
Proposition 24: Let be a scheme, an open subscheme, and the complementary closed subscheme. Let be an abelian sheaf on . Then, there is a spectral sequence
Let us note that, in particular,
Thus, we see that this spectral sequence, in particular, gives us a measure on how far is from just the naive object .
Excision and purity
One intuition one might garner from thinking of as the ‘cohomology of with supports in ‘ is that this cohomology group perhaps shouldn’t change when one shrinks the space around . One can also roughly intuit this from Proposition 23 since if we shrink around we will also shrink and so the cokernel and kernels of the maps from should be insensitive to this simultaneous shrinking.
The excision theorem, which we presently state, is precisely the rigorous version of this intuition:
Proposition 25 (the Excision Theorem): Let be a scheme, an open subscheme, and the complementary closed subscheme. Suppose that is an etale morphism with the property that is an isomorphism. Then, there is a natural isomorphism
for all .
Proof: See [Mil, Proposition III.1.27].
One useful corollary of this comes from considering what it says when is just a point . Namely, for every pointed etale map (an etale neighborhood of ), such that is the only point in the fiber over and is an isomorphism, the excision theorem tells us that
We might then try to pass to the limit over all the etale neighborhoods of (i.e. ‘shrink the neighborhood to zero’) and hope that the limiting structure is more concrete.
This, in fact, can be done but only with care:
Lemma 26: Let be a Noetherian scheme and let be a closed point of . Then, there is a natural isomorphism
where is the henselization of and is the obvious map.
Proof: Using Proposition 23 and the Excision Theorem one reduces to showing this as a statement roughly showing that there exists a cofinal sequence of etale neighborhoods of such that both and are quasi-compact. Indeed, once one knows this one obtains the result by combining the Excision Theorem, Proposition 23, [Stacks, Tag09YQ] , the definition of Henselization (e.g. see [Fu, Page 102] and [Stacks,05KS]), and [Stacks, Tag01YX]. The reason that such a sequence exists is that one can consider only affine neighborhoods (for which ) and for such we have that is Noetherian since is etale (and so is locally Noetherian) and compact. So then, not only is quasi-compact but so then is .
This focusing on Noetherianity assumptions may seem a bit dramatic/pedantic now, but it turns out to be a pivotal difference between schemes and adic spaces — in the former category Noetherianity conditions cover essentially all reasonable examples, and in the latter this is far from true.
Regardless, the excision lemma (and by extension Lemma 26) allow one to often times reduce questions about cohomology with supports to much more familiar ones. Namely, the fact that cohomology with supports is insensitive to shrinking of etale neighborhoods and the fact that every smooth scheme etale locally looks like affine space makes one wonder whether etale cohomology with supports can be reduced to a calculation of just affine space.
The key, simple to prove, observation necessary to make this precise is the following:
Lemma 27: Let be a scheme, a smooth scheme of relative dimension , and a closed subscheme (fiberwise) pure of codimension , such that is smooth. Then, for every point there exists a neighborhood of and an etale map such that .
Proof: See [SGA 1, Théorème II.4.10].
Let us say that a pair as in Lemma 27 is a smooth pair of codimension over . Then, one can interpret Lemma 27 as saying that every smooth pair of codimension over etale locally looks like . Thus, by etale local nature of one can generally reduce abstract computations to ones of the form which can (hopefully) be explicitly computed)
It is precisely by this reduction to specific this specific case that the following purity result is proven:
Proposition 28 (the Purity Theorem): Let be a smooth pair of codimension over and let be a locally constant torsion sheaf whose stalks have order invertible in . Then,
Proof: This is [Mil, Theorem IV.5.1] and/or [Fu, Corollary 8.5.6]. The proof is, as we indicated above, a reduction to the affine case and then an explicit computation.
Remark 29: The kind of explicit computation needed in the above is, essentially, the one done in this post, at least in the case when .
As a corollary of the Purity Theorem and Proposition 24 we immediately get a more concrete statement:
Corollary 30: Let be a field and let a smooth -variety of dimension and a smooth closed -variety of pure codimension . Then there is an isomorphism of -modules
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