A computation a day: compactly supported cohomology

In this post we compute the compactly supported cohomology of some simple varieties.

A reminder on some definitions

Before we dive into the calculation, let us recall the definition of the compactly supported cohomology groups H^i_c(X,\mathbb{Q}_\ell) for a smooth integral variety X/k, where k is any algebraically closed field who, just for convenience, satisfies \mathrm{char}(k)\ne \ell. To extend the definition to k not algebraically closed, use the definition below for the variety base changed to \overline{k} (where it will then have a Galois action!).

The fancy ‘adic’ definition would be just to define H^i_c(X,\mathbb{Q}_\ell) to be the ‘adic’ sheaf Rf_! \mathbb{Z}_\ell. Here f:X\to\mathrm{Spec}(k) is the structure morphism, \mathbb{Z}_\ell is the ‘adic’ sheaf \left\{\mathbb{Z}/\ell^n\mathbb{Z}\right\}, and f_! is the usual ‘shriek forward’, a member of the Six Operations Gang. For this perspective, see these notes of Conrad. We could then just define H^i_c(X,\mathbb{Q}_\ell) to be R f_!\mathbb{Q}_\ell, in this formalism, which amounts to tensoring Rf_! \mathbb{Z}_\ell‘s inverse limit with \mathbb{Q}_\ell.

A more down-to-earth definition of the compactly supported cohomology is as follows:

H^i_c(X,\mathbb{Q}_\ell):=\left(\varprojlim H^i_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\right)\otimes_{\mathbb{Z}_\ell}\mathbb{Q}_\ell

which reduces us to defining the compactly supported (étale) cohomology of a torsion sheaf \Lambda on X_\mathrm{et}. We then define this as follows. Let f:X\to\mathrm{Spec}(k) be the structure morphism of X. Then, by Nagata compactification we can (densely) open embed j:X\hookrightarrow \overline{X}. We then define H^i_c(X,\Lambda) as follows:

H^i_c(X,\Lambda):= H^i_\mathrm{et}(\overline{X},j_!\Lambda)

One can show that this definition is independent of compactification (essentially following from base change), and satisfies sufficiently nice properties–see Milne’s Étale Cohomology, Chapter VI, section 3.

Remark: The rough intuition for H^i_c is as follows. One often times thinks of H^i_\mathrm{et} as being an analogy of singular cohomology, as codified by Grothendieck’s comparison theorem (see this nice note). Then, under this analogy H^i_c corresponds to the usual compactly supported singular cohomology from topology, as intuited in, for example, Bott and Tu’s Differential Forms in Algebraic Topology. It’s just the cohomology of compactly supported cochains.

Remark: Note that the compactly supported cohomology of something proper is just usual cohomology. More rigorously, if X/k is proper, then H^i_c(X,\mathbb{Q}_\ell) agrees with H^i(X,\mathbb{Q}_\ell)

A useful tool

Before we begin actually computing the compactly supported cohomology of some simple spaces, we first recall a simple fact which will, in fact, reduce the computation of H^i_c to previously known quantities. For the remainder of this post, let us define the sheaf \Lambda_{m,S} on S_\mathrm{et}, for any variety S/k, to be the constant sheaf \mathbb{Z}/n\mathbb{Z}. This will be useful as we will be considering the constant sheaf \mathbb{Z}/m\mathbb{Z} on various spaces.

NB: All cohomologies unadorned with a subscript should be assumed to be étale and/or limit of étale (i.e. \ell-adic cohomology).

This tool comes from the very simple question: given a compactification \overline{X}, how does H^i_c(X,\mathbb{Q}_\ell) relate to H^i(\overline{X},\mathbb{Q}_\ell)? Namely, how close are the cohomologies, on \overline{X}, of j_!\Lambda_{\ell^n,X} and \Lambda_{\ell^n,\overline{X}}? Intuitively, through some sort of excisions-esque thought process, one would expect the difference to depend on the complementary closed subscheme i:Z\hookrightarrow X (i.e. X-U with the reduced substructure).

This intuition, as it turns out, is entirely correct. Namely, we have the following nice sequence which connect the three objects:

0\to j_!\Lambda_{\ell^n,X}\to \Lambda_{\ell^n,\overline{X}}\to i_\ast\Lambda_{\ell^n,Z}\to 0

 where this is a sequence of sheaves on \overline{X}. This sequence can be found, for example, in Milne’s Lectures on Étale Cohomology (i.e. the notes, not the book)  in the proof of Proposition 18.3(b) under the guise of a ‘push-pull’ sequence (take \mathcal{F}=\Lambda_{\ell^n,X} and use the fact that j is a dense open embedding).

From this sequence we then get a long exact sequence in cohomology:

\begin{aligned} 0\to & H^0(\overline{X},j_!\Lambda_{\ell^n,X}) \to H^0(\overline{X},\Lambda_{\ell^n,\overline{X}})\to H^0(\overline{X},i_\ast\Lambda_{\ell^n,Z})\to \\ & H^1(\overline{X},j_!\Lambda_{\ell^n,X})\to H^1(\overline{X},\Lambda_{\ell^n,\overline{X}})\to H^1(\overline{X},i_\ast\Lambda_{\ell^n,\overline{Z}})\to \cdots\end{aligned}

But, using the definition of compactly supported cohomology, and the fact that i is a closed embedding (so R^j i_\ast=0 for j>0) we can rewrite this as

\begin{aligned} 0\to & H^0_c(X,\Lambda_{\ell^n,X}) \to H^0(\overline{X},\Lambda_{\ell^n,\overline{X}})\to H^0(Z,\Lambda_{\ell^n,Z})\to \\ & H^1_c(X,\Lambda_{\ell^n,X})\to H^1(\overline{X},\Lambda_{\ell^n,\overline{X}})\to H^1(Z,\Lambda_{\ell^n,\overline{Z}})\to \cdots\end{aligned}

which gives me the desired connection between compactly supported cohomology of X, and cohomology of \overline{X}.

Affine Space

We begin by computing the compactly supported cohomology of the simplest non-proper variety: affine space \mathbb{A}^m_k. Now, by the last section we get a long exact sequence of cohomology groups

\begin{aligned} 0\to & H^0_c(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z}) \to H^0(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^0(\mathbb{P}^{m-1}_k,\mathbb{Z}/\ell^n\mathbb{Z})\to \\ & H^1_c(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(\mathbb{P}^{m-1},\mathbb{Z}/\ell^n\mathbb{Z})\to \cdots \\ & H^{2m}_c(\mathbb{A}^m_k,\mathbb{Z}_\ell^n\mathbb{Z})\to H^{2m}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z}) \to H^{2m}(\mathbb{P}^{m-1}_k,\mathbb{Z}/\ell^n\mathbb{Z})\to 0 \end{aligned}

Now, we have previously computed the cohomology of projective space:

H^i(\mathbb{P}^m,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=0,2,\ldots,2m\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

But, also, since j:\mathbb{A}^m_k\hookrightarrow\mathbb{P}^m_k is an open embedding, it’s trivial that

0=H^0(\mathbb{P}^m_k,j_!\Lambda_{\ell^m,\mathbb{A}^n_k})=H^0_c(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})

Thus, we immediately deduce that for all i=1,\ldots,m-1 we have a short exact sequence

H^{2i}(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\hookrightarrow H^{2i}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^{2i}(\mathbb{P}^{m-1}_k,\mathbb{Z}/\ell^n\mathbb{Z})\twoheadrightarrow H^{2i+1}(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})

For i=2m we get the short exact sequence

0\to H^{2m}(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^{2m}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})

from which we immediately deduce that H^{2m}(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})=\mathbb{Z}/\ell^n\mathbb{Z}).

But, one can easily check, from the definition (and the explicit computation of the cohomology of \mathbb{P}^m_k) that the map

H^{2i}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^{2i}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})

is an isomorphism (this is clear if one thinks about the isomorphism in terms of the cycle class map). Thus, we deduce the following

H^i_c(\mathbb{A}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=2m\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

So, by passing to the limit (taking for granted that the transition maps are multiplication by \ell) we deduce that

H^i_c(\mathbb{A}^m_k,\mathbb{Q}_\ell)=\begin{cases}\mathbb{Q}_\ell & \mbox{if}\quad i=2m\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

which is what we’d topologically expect.

Remark: If any of the above is unsavory to the reader (for example why the map H^{2i}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})\to H^{2i}(\mathbb{P}^m_k,\mathbb{Z}/\ell^n\mathbb{Z})) is an isomorphism, one could alternatively proceed as follows. Do the even easier computation of H^i_c(\mathbb{A}^1_k,\mathbb{Q}_\ell) (here there the last parenthetical statement is vacuous), and use the fact that H^i_c satisfies the Kunneth formula.

Smooth affine curves

We now compute H^i_c(X,\mathbb{Q}_\ell) where X is a smooth affine curve over k. In this case, once again analogizing to the topological case, we’d hope that H^1_c(X,\mathbb{Q}_\ell), the only non-trivial group, will pick up on the genus of its (unique!) smooth compactification \overline{X}, and the number of ‘holes’ left by X in \overline{X} (i.e. the size of \overline{X}-X.

We proceed as above. Namely, let j:X\hookrightarrow\overline{X} be a compactification of X, where \overline{X} is smooth and integral (there is a unique such curve!). We then write down the long exact sequence we’ve been continually using:

\begin{aligned} 0\to & H^0_c(X,\mathbb{Z}/\ell^n\mathbb{Z}) \to H^0(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to H^0(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to \\ & H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to \\ & H^{2}_c(X,\mathbb{Z}_\ell^n\mathbb{Z})\to H^{2}(X,\mathbb{Z}/\ell^n\mathbb{Z}) \to H^{2}(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to 0 \end{aligned}

where Z=\overline{X}-X with the reduced structure. Note that since Z is zero dimensional, all of its higher cohomology vanishes. This, together with the computation H^0_c(X,\mathbb{Z}/\ell^n\mathbb{Z})=0 (for the same reason as in the previous case), the above simplifies to the following two short exact sequences

H^0(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\hookrightarrow H^0(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\twoheadrightarrow H^1(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})

and

0\to H^2_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^2(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to 0

But, as we’ve previously computed, the cohomology of a smooth projective curve C of genus g is

H^i(C,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}\ell^n\mathbb{Z} & \mbox{if}\quad i=0,2\\ (\mathbb{Z}/\ell^n\mathbb{Z})^{2g} & \mbox{if}\quad i=1\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

Moreover, H^0(Z,\mathbb{Z}/\ell^n\mathbb{Z})=(\mathbb{Z}/\ell^n\mathbb{Z})^{s}, where s is the number of points in Z. Thus, the above simplifies to an exact sequence

0\to\mathbb{Z}/\ell^n\mathbb{Z}\to (\mathbb{Z}/\ell^n\mathbb{Z})^s\to H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to (\mathbb{Z}/\ell^n\mathbb{Z})^{2g}\to 0

That said, we know precisely what the map \mathbb{Z}/\ell^n\mathbb{Z}\to (\mathbb{Z}/\ell^n\mathbb{Z})^s is–it’s just the diagonal map! Thus, we actually get a short exact sequence

0\to (\mathbb{Z}/\ell^n\mathbb{Z})^{s-1}\to H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to (\mathbb{Z}/\ell^n\mathbb{Z})^{2g}\to 0

Thus, we know that H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z}) is a \mathbb{Z}/\ell^n\mathbb{Z}-module of size (\ell^n)^{2g+s-1}. Using this, together with the fact that H^1_c is self-dual (by Poincaré duality!) we thus obtain the following:

H^i_c(X,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}(\mathbb{Z}/\ell^n\mathbb{Z})^{2g+s-1} & \mbox{if}\quad i=1\\ \mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=2\\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

Then, by passing to the limit (taking for granted that the transition maps are what we want) we thus conclude that:

H^i_c(X,\mathbb{Q}_\ell)=\begin{cases}\mathbb{Q}_\ell^{2g+s-1} & \mbox{if}\quad i=1\\ \mathbb{Q}_\ell & \mbox{if}\quad i=2\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

Remark: Note that this implies something interesting about curves. Namely, we already knew that for a curve X, with smooth compactification \overline{X}, the genus of \overline{X} was an isomorphism invariant of X. The above shows that, even further, how many points are in \overline{X}-X is independent of embedding, and is also an isomorphism invariant.

Multiply Punctured Projective Space

The following thought-experiment always seemed like a very natural presentation of the need for algebraic topology to a young math major. Suppose that such a student has taken some introduction to topology, meaning something like just point-set. They are very good at telling spaces apart. You might hand them two spaces X and Y and ask “are they homeomorphic?” They might then go “of course not, X is compact and Y is not”, or “of course not, X is locally path connected and Y is not”, or even “of course not, X is metrizable and Y isn’t even T_4!”

That said, hand them the spaces S^2 and S^1\times S^1 (the torus), and they’ll be surprisingly mute. Pretty much all of the basic invariants one learns in a first course in point-set topology are useless in telling these two spaces apart. They are both compact. They are both path connected. They are both metrizable. The list goes on. That said, it is obvious that they are not geometrically the same–one has ‘a hole’! And thus is born a clear and present need for some sort of algebraic formalism to measure ‘higher connectedness’–it’s just the next logical invariant after separation axioms and their brethren.

I think that in algebraic geometry, the following question plays a similar role: suppose that s\ne t, are the k-varieties \mathbb{P}^m_k-\{x_1,\ldots,x_s\} and \mathbb{P}^m_k-\{y_1,\ldots,y_t\}, for k-points x_i and y_j, isomorphic? One can quickly try and use the basic invariants one learns in a first course in cohomology to tell them apart, but to no avail. Neither are proper/projective (the obvious map to \mathbb{P}_k^m is not closed). Assuming m>1, neither are affine (they have the same global sections as \mathbb{P}^m_k by ‘Algebraic Hartog’s Lemma’). All of the properties that are preserved under birationality are preserved (e.g. dimension). One could try and compute coherent cohomology, but the only things would could argue would have to be ‘functorial sheaves’ (i.e. pairs of sheaves \mathcal{F},\mathcal{G} on each of the two spaces such that for any isomorphism f between them f^\ast\mathcal{F}=\mathcal{G}, such as the structure sheaf or the contangent sheaf). And, unless, I have miscalculated, these two agree.

We are thus in a similar situation to the second paragraph. Namely, we know that these two spaces ‘shouldn’t’ be isomorphic (the amount of points removed seems like an invariant!), but we’re at a loss as to how and prove this. We need some sort of more sophisticated tool. Of course, our first intuition is that they aren’t homeomorphic over \mathbb{C}–they have different singular (co)homologies! Whenever we have this intuition, a little lightbulb in our head should go off that to make this rigorous for general varieties we need only use some sort of Weil cohomology theory. In particular, compactly supported cohomology fits the bill!

So, let’s compute H^i_c(X,\mathbb{Q}_\ell) where X=\mathbb{P}^m_k-\{x_1,\ldots,x_s\}, where the x_i are k-points, and \ell\ne\mathrm{char}(k).

As per usual, we begin by finding an ‘obvious’ compactification of X. In our case, it is literally handed to us. Namely, let’s let \overline{X}:=\mathbb{P}^m_k, and let j:X\hookrightarrow\overline{X} be the obvious open embedding. Let then, as usual Z=\overline{X}-X, with the reduced subscheme structure, which in this case is just the disjoint union of s points. Then, by applying our savior-sequence from the second section we get the following exact sequence

\begin{aligned} 0\to & H^0_c(X,\mathbb{Z}/\ell^n\mathbb{Z}) \to H^0(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to H^0(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to \\ & H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to H^1(Z\mathbb{Z}/\ell^n\mathbb{Z})\to \cdots \\ & H^{2m}_c(X,\mathbb{Z}_\ell^n\mathbb{Z})\to H^{2m}(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z}) \to H^{2m}(Z,\mathbb{Z}/\ell^n\mathbb{Z})\to 0 \end{aligned}

Now, we simplify this by replacing the obvious pieces. Namely, H^0_c(X,\mathbb{Z}/\ell^n\mathbb{Z})=0, we know the cohomology of \mathbb{P}^m, and the cohomology of a disjoint union of points is also obvious. Thus, we obtain the following sequences

0\to \mathbb{Z}/\ell^n\mathbb{Z}\to (\mathbb{Z}/\ell^n\mathbb{Z})^m\to H^1_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to 0

and for i>1

0\to H^i_c(X,\mathbb{Z}/\ell^n\mathbb{Z})\to H^i(\overline{X},\mathbb{Z}/\ell^n\mathbb{Z})\to 0

But, we know precisely what the map \mathbb{Z}/\ell^n\mathbb{Z}\to (\mathbb{Z}/\ell^n\mathbb{Z})^m is! It’s just the map x\mapsto (x,\ldots,x). Thus, from this, and the second sequence above we obtain the following:

H^i_c(\mathbb{P}^m_k-\{x_1,\ldots,x_s\},\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=2,\ldots,2m\\ (\mathbb{Z}/\ell^n\mathbb{Z})^{s-1} & \mbox{if}\quad i=1\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

Then, passing to the limit (assuming, as per usual, that the transition maps are what we expect) we find the following:

H^i_c(\mathbb{P}^m_k-\{x_1,\ldots,x_s\},\mathbb{Q}_\ell)=\begin{cases}\mathbb{Q}_\ell & \mbox{if}\quad i=2,\ldots,2m\\ \mathbb{Q}_\ell^{s-1} & \mbox{if}\quad i=1\\ 0 & \mbox{if}\quad \mathrm{otherwise}\end{cases}

Thus, we can tell apart, for s,t>0 (if either is zero, we can think about projectiveness!) \mathbb{P}^m_k-\{x_1,\ldots,x_s\} and \mathbb{P}^m_k-\{y_1,\ldots,y_t\} by looking at their cohomology.

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