Punctured disks and punctured curves and cohomology, oh my! (Part II: cohomology with supports)

This is a continuation of this post.

Cohomology with supports

Desiderata

We now wish to move on from the proper case of Proposition 2 to the case of non-proper smooth connected curves U. 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 U by embedding it into a proper smooth connected curve X and somehow understand what the ‘difference’ between the cohomology of U and X is.

Namely, we want a way to explicitly compare the cohomology of a scheme X and an open subset U. Our intuition strongly pushes us to imagine that somehow their should be a relationship involving the cohomology for X, U, and Z:=X-U (the closed complement of U). 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 X, U, and Z are related. Namely, we have the following simple theorem:

Observation 21: Let k be an algebraically closed field and let X be proper variety over k, U and open subscheme, and Z its complement. Then, for each prime \ell and each n\geqslant 1 there is a long exact sequence in compactly supported cohomology

0\to\cdots H^i_{c,\mathrm{et}}(U,\mathbb{Z}/\ell^n\mathbb{Z})\to H^i_{c,\mathrm{et}}(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^i_{c,\mathrm{et}}(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to\cdots

Proof: For example one can see [Mil, Remark III.1.30]. \blacksquare

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 Z is a closed subset of a scheme X then essentially all the scheme structures on Z have the same underlying reduced scheme. But, the passage from Z to Z_\mathrm{red} 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 U and Z both in to X but maps on cohomology in different directions. This phenomenon cannot occur for normal cohomology which is always contravariantly functorial. So, the open embedding U\hookrightarrow X gives rise to a map

H^i_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^i_\mathrm{et}(U,\mathbb{Z}/\ell^n\mathbb{Z})

and while it wouldn’t make sense for the kernel of this map to be H^i_\mathrm{et}(Z,\mathbb{Z}/\ell^n\mathbb{Z}) it could, hopefully, still be some cohomology group H^i_{Z,\mathrm{et}} related to Z.

In particular, we already know what the answer should be at least for H^0_{Z,\mathrm{et}}. Namely, for an arbitrary sheaf we can certainly set

H^0_{Z,\mathrm{et}}(X,\mathcal{F}):=\ker(H^0_\mathrm{et}(X,\mathcal{F})\to H^0_\mathrm{et}(U,\mathcal{F}))

In words one should think of H^0_{Z,\mathrm{et}}(X,\mathcal{F}) as the ‘sections of \mathcal{F}(X) with supports in Z‘. One can check that the functor

\mathsf{Ab}(X_\mathrm{et})\to \mathsf{Ab}:\mathcal{F}\mapsto H^0_{Z,\mathrm{et}}(X,\mathcal{F})

is left exact and we define H^i_{Z,\mathrm{et}}(X,-) to be its i^\text{th}-derived functor. We call this the cohomology with supports in Z.

As promised, and essentially as constructed, we have the following:

Proposition 23: Let X be a scheme, U an open subscheme, and Z its closed complement. Then, for any abelian sheaf \mathcal{F} on X_\mathrm{et} there is a long exact sequence of abelian groups

\cdots \to H^i_{Z,\mathrm{et}}(X,\mathcal{F})\to H^i_\mathrm{et}(X,\mathcal{F})\to H^i_\mathrm{et}(U,\mathcal{F})\to \cdots

Proof: This is [Mil, Proposition III.1.25] or [Fu, Proposition 5.6.11]. \blacksquare

The last thing we mention in this subsection is that there is a natural way to interpret the group H^1_{Z,\mathrm{et}}(X,\mathcal{F}) in terms of the theory of torsors (e.g. see the notes from this post). Namely, one can show that

H^1_{Z,\mathrm{et}}(X,\mathcal{F})\cong \left\{(\mathcal{G},\varphi):\begin{aligned}(1)&\quad \mathcal{G}\text{ is an }\mathcal{F}\text{-torsor}\\ (2)&\quad \varphi:\mathcal{G}_U\to\mathcal{F}_U\text{ is an isomorphism}\end{aligned}\right\}/\text{iso.}

Here two pairs (\mathcal{G},\varphi) and (\mathcal{G}',\varphi') are isomorphic if there is an isomorphism \psi:\mathcal{G}\xrightarrow{\approx}\mathcal{G}' so that \varphi=\varphi'\circ\psi\mid_U. With this interpretation the map

H^0_{\mathrm{et}}(U,\mathcal{F})\to H^1_{Z,\mathrm{et}}(X,\mathcal{F})

can be interpreted as sending g\in\mathcal{F}(U) to the pair (\mathcal{F},\varphi_g) where \varphi_g:\mathcal{F}\mid_U\to\mathcal{F}\mid_U is the multiplication by g map. Similarly, the map

H^1_{Z,\mathrm{et}}(X,\mathcal{F})\to H^1_{\mathrm{et}}(X,\mathcal{F})

can be thought of as the map which associates to a pair (\mathcal{G},\varphi) the torsor \mathcal{G}.

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 H^0_{Z,\mathrm{et}}(X,\mathcal{F}) as follows:

H^0_{Z,\mathrm{et}}(X,\mathcal{F}):=\ker(\mathcal{F}\to j_\ast j^\ast \mathcal{F})(X)

one then might wonder what the relationship between cohomology with supports and the sheaf \mathscr{K}:=\ker(\mathcal{F}\to j_\ast j^\ast\mathcal{F}).

In particular, an incredibly naive guess one might have is that H^i_{Z,\mathrm{et}}(X,\mathcal{F}) is the cohomology of \mathscr{K}, but this is totally wrong. For example, note that, by definition we have an exact sequence

0\to\mathscr{K}\to \mathcal{F}\to j_\ast j^\ast\mathcal{F}\to 0

and so we get a long exact sequence that looks like

\cdots \to H^i_{\mathrm{et}}(X,\mathscr{K})\to H^i_\mathrm{et}(X,\mathcal{F})\to H^i_\mathrm{et}(X,j_\ast j^\ast\mathcal{F})\to \cdots

and if we want this to match Proposition 23 we’d want H^i_\mathrm{et}(X,j_\ast j^\ast\mathcal{F}) to match H^i_\mathrm{et}(U,\mathcal{F}) but this is horribly, horribly wrong. For example, in the case where U is an affine open subcurve of the proper connected smooth curve X then one has that j_\ast j^\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}}=\underline{\mathbb{Z}/\ell^n\mathbb{Z}} and so

H^i_{\mathrm{et}}(X,j^\ast j_\ast\underline{\mathbb{Z}/\ell^n\mathbb{Z}})=H^i_{\mathrm{et}}(X,\mathbb{Z}/\ell^n\mathbb{Z})

which, as Proposition 2 tells us, is quite different from H^i_{\mathrm{et}}(U,\mathbb{Z}/\ell^n\mathbb{Z}) in general.

That said, there is a relationship between cohomology, \mathscr{K}, and H^i_{Z,\mathrm{et}}(X,\mathcal{F}). Namely, let us set i^!:= i^\ast\mathscr{K} and call it the shriek back of \mathcal{F} along i. This scary terminology comes from the fact that (i_\ast,i^!) is an adjoint pair (e.g. see [Fu, Proposition 5.4.2]) where, in general, for a map f we define f^! so that (f_\ast,f^!) is an adjoint pair.  One can show that i^! is left exact (e.g. see loc. cit.) and thus we have a well-defined notion of the right-defined functors

R^i i^!:\mathsf{Ab}(X_\mathrm{et})\to \mathsf{Ab}(Z_\mathrm{et})

which are often, somewhat confusingly, written as \underline{H}^i_{Z,\mathrm{et}}(X,\mathcal{F}). In particular, note that \underline{H}^i_{Z,\mathrm{et}}(X,\mathcal{F}) is a sheaf on Z.

Now, as one can check one has that i_\ast i^!\mathcal{F}=\mathscr{K} and so, in particular, we see that

(i^! \mathcal{F})(Z)=(i_\ast i^! \mathcal{F})(X)=\mathscr{K}(X)=H^0_{Z,\mathrm{et}}(X,\mathcal{F})

so we see that the composition

\mathsf{Ab}(X_\mathrm{et})\xrightarrow{i^!}\mathsf{Ab}(Z_\mathrm{et})\xrightarrow{\Gamma(Z,-)}\mathsf{Ab}

is precisely H^0_{Z,\mathrm{et}}(X,-). In particular since (as one can check) i^! preserves injectives we obtain the following from the Grothendieck spectral sequence:

Proposition 24: Let X be a scheme, U an open subscheme, and Z the complementary closed subscheme. Let \mathcal{F} be an abelian sheaf on X_\mathrm{et}. Then, there is a spectral sequence

E_2^{p,q}=H^p_\mathrm{et}(Z,\underline{H}^q_{Z,\mathrm{et}}(X,\mathcal{F}))\implies H^{p+q}_{Z,\mathrm{et}}(X,\mathcal{F})

Let us note that, in particular,

\begin{aligned}H^p_\mathrm{et}(Z,\underline{H}^0_{Z,\mathrm{et}}(X,\mathcal{F})) &=H^p_\mathrm{et}(Z,i^!\mathcal{F})\\ &=H^p_\mathrm{et}(X,i_\ast i^!\mathcal{F})\\ &\cong H^p_\mathrm{et}(X,\mathscr{K})\end{aligned}

Thus, we see that this spectral sequence, in particular, gives us a measure on how far H^p_{Z,\mathrm{et}}(X,\mathcal{F}) is from just the naive object H^p_\mathrm{et}(X,\mathscr{K}).

Excision and purity

One intuition one might garner from thinking of H^i_{Z,\mathrm{et}}(X,\mathcal{F}) as the ‘cohomology of \mathcal{F} with supports in Z‘ is that this cohomology group perhaps shouldn’t change when one shrinks the space X around Z. One can also roughly intuit this from Proposition 23 since if we shrink X around Z we will also shrink U and so the cokernel and kernels of the maps from H^i_{\mathrm{et}}(X,\mathcal{F})\to H^i_{\mathrm{et}}(U,\mathcal{F}) 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 X be a scheme, U an open subscheme, and Z the complementary closed subscheme. Suppose that f:X'\to X is an etale morphism with the property that f:f^{-1}(Z)\to Z is an isomorphism. Then, there is a natural isomorphism

H^i_{Z,\mathrm{et}}(X,\mathcal{F})\xrightarrow{\approx}H^i_{f^{-1}(Z),\mathrm{et}}(X',f^{-1}\mathcal{F})

for all i\geqslant 0.

Proof: See [Mil, Proposition III.1.27]. \blacksquare

One useful corollary of this comes from considering what it says when Z is just a point z. Namely, for every pointed etale map (U,u)\to (Z,z) (an etale neighborhood of (Z,z)), such that u is the only point in the fiber over z and k(z)\to k(u) is an isomorphism, the excision theorem tells us that

H^i_{z,\mathrm{et}}(X,\mathcal{F})\xrightarrow{\approx}H^i_{u,\mathrm{et}}(U,f^{-1}\mathcal{F})

We might then try to pass to the limit over all the etale neighborhoods of (Z,z) (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 X be a Noetherian scheme and let z be a closed point of X. Then, there is a natural isomorphism

H^i_{z,\mathrm{et}}(X,\mathcal{F})\xrightarrow{\approx} H^i_{z,\mathrm{et}}(\mathrm{Spec}(\mathcal{O}_{X,z}^h), f^{-1}\mathcal{F})

where \mathcal{O}_{X,z}^h is the henselization of \mathcal{O}_{X,z} and f:\mathrm{Spec}(\mathcal{O}_{X,z}^h)\to X 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 (V_\alpha,v_\alpha) lf (X,z) such that both V_\alpha and V_\alpha-v_\alpha 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 (V_\alpha,v_\alpha) (for which f^{-1}(z)\cong v_\alpha) and for such V_\alpha we have that V_\alpha is Noetherian since V_\alpha\to X is etale (and so V_\alpha is locally Noetherian) and compact. So then, not only is V_\alpha quasi-compact but so then is V_\alpha-\{v_\alpha\}. \blacksquare

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 S be a scheme, X\to S a smooth scheme of relative dimension n, and Z\hookrightarrow X a closed subscheme (fiberwise) pure of codimension c, such that Z\to S is smooth. Then, for every point x\in X there exists a neighborhood U of x and an etale map f:U\to\mathbb{A}^n_S such that U\cap Z=f^{-1}(\mathbb{A}^{n-c}_S).

Proof: See [SGA 1, Théorème II.4.10]. \blacksquare

Let us say that a pair (X,Z) as in Lemma 27 is a smooth pair of codimension c over S. Then, one can interpret Lemma 27 as saying that every smooth pair of codimension c over s etale locally looks like (\mathbb{A}^n_S,\mathbb{A}^{n-c}_S). Thus, by etale local nature of H^i_{Z,\mathrm{et}}(X,\mathcal{F}) one can generally reduce abstract computations to ones of the form H^i_{\mathbb{A}^{n-c}_S,\mathrm{et}}(\mathbb{A}^n_S,\mathcal{F}) 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 (X,Z) be a smooth pair of codimension c over S and let \mathcal{F} be a locally constant torsion sheaf whose stalks have order invertible in \mathcal{O}_S(S). Then,

\underline{H}_{Z,\mathrm{et}}(X,\mathcal{F})\cong\begin{cases}i^\ast\mathcal{F}(-c) & \mbox{if}\quad i=2c\\ 0 & \mbox{if}\quad \text{otherwise}\end{cases}

 

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. \blacksquare

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 c=n.

As a corollary of the Purity Theorem and Proposition 24 we immediately get a more concrete statement:

Corollary 30: Let k be a field and let X a smooth k-variety of dimension n and Z a smooth closed k-variety of pure codimension c. Then there is an isomorphism of \mathrm{Gal}(k^{\mathrm{sep}}/k)-modules

H^i_{Z_{k^\mathrm{sep}},\mathrm{et}}(X_{k^\mathrm{sep}},\mathcal{F}_{k^\mathrm{sep}})\cong \begin{cases}0 & \mbox{if}\quad i<2c\\ H^{i-2c}_\mathrm{et}(Z_{k^\mathrm{sep}},i^\ast\mathcal{F}_{k^\mathrm{sep}})(-c) & \mbox{if}\quad i\geqslant 2c\end{cases}

References

[Con] Conrad, B., 2015. Abelian varietieshttps://stuff.mit.edu/people/fengt/249C.pdf

[EP] Engler, A.J. and Prestel, A., 2005. Valued fields. Springer Science & Business Media.

[Fu] Fu, L., 2011. Etale cohomology theory (Vol. 13). World Scientific.

[Har] Hartshorne, R., 2013. Algebraic geometry (Vol. 52). Springer Science & Business Media.

[Lit] Litt, Daniel. Cohomology of curves. https://www.daniellitt.com/s/cohomologyofcurves.pdf

[Mac] Mack-Crane, Sander. Normalization of Algebraic and Arithmetic Curveshttps://algebrateahousejmath.wordpress.com/2016/11/27/normalization-of-algebraic-and-arithmetic-curves/

[Mil] Milne, J.S. and Milne, J.S., 1980. Etale cohomology (PMS-33) (No. 33). Princeton university press.

[Nag] Nagata, M., 1962. Local rings. Interscience Tracts in Pure and Appl. Math.

[Ols] Olsson, M., 2016. Algebraic spaces and stacks (Vol. 62). American Mathematical Soc.

[Poo] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc.

[SGA 1] Grothendieck, A. and Raynaud, M., 2002. Revêtementset groupe fondamental (SGA 1). arXiv preprint math/0206203.

[Stacks] The Stacks Project Authors, The Stacks Project, https://stacks.math.columbia.edu

[Star] Starr, Jason. Brauer groups and Galois cohomology of function
fields of varieties
. https://www.math.stonybrook.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf

[Sut] Sutherland, Andrew. Totally ramified extensions and Krasner’s lemmahttp://math.mit.edu/classes/18.785/2016fa/LectureNotes11.pdf

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