Concrete example of calculation of $\ell$-adic cohomology
Let $p$ and $\ell$ be distinct prime numbers.
Consider in the affine plane $\mathbb{A}^2_{\mathbb{F}_p}$ with coordinates $(x,y)$ the union $L$ of the axes $x = 0$ and $y = 0$.
How does one compute the $\ell$-adic cohomology groups with compact support $H^i_c(L,\mathbb{Q}_\ell)$? I thought I had some idea of what $\ell$-adic cohomology is, but I don't even manage to do this example...
Solution 1:
(1) All we really need to know to make the computation is that cohomology with compact support $H^*_c$ (with values in $K$-vector spaces) satisfies the following properties:
- (Localization sequence) If $i: Z \hookrightarrow X$ is a closed immersion and $j:U\hookrightarrow X$ the complementary open immersion, then for any sheaf $F$ we have a long exact sequence $$ \to H^{i}_c(U) \to H^{i}_c(X) \to H^{i}_c(Z) \to H^{i+1}_c(U) \to $$
- (Cohomology of the affine space) $ H^{i}_c(\mathbb{A}^n) = \begin{cases} K & if~ i=2n \cr 0 & else \end{cases} $
(2) Now we can play with the long exact sequences
Writing the localization sequence for $\mathbb{A}^1 = \{0\} \coprod \mathbb{G}_m$ one finds $ H^{i}_c(\mathbb{G}_m) = \begin{cases} K & if~ i=1,2 \cr 0 & else \end{cases} $ This is dual to $H^{2-i}(\mathbb{G}_m)$ as expected.
Then the localization sequence for $L = \mathbb{A}^1 \coprod \mathbb{G}_m$ reduces to
$$
0\to H^{0}_c(L) \to 0 \to
K \to H^{1}_c(L) \to 0 \to
K \to H^{2}_c(L) \to K \to
0
$$
so
$
H^{i}_c(L) = \begin{cases} K & if~ i=1 \cr K^2 & if~ i=2 \cr 0 & else \end{cases}
$
(3) I should add a few word about the localizing sequence. For any immersion $j: U\hookrightarrow X$, we have 3 functors on sheaves
- the restriction functor $j^*$
- its right adjoint: the classical direct image $j_*$
- the extension by zero $j_!$.
Facts (see J.S. Milne's lecture notes for example):
- $j_!$ is always exact
- If $j$ is a closed immersion then $j_! = j_*$.
- If $i: Z \hookrightarrow X$ is a closed immersion and $j:U\hookrightarrow X$ the complementary open immersion, then for any sheaf $F$ we have an exact sequence $$ 0 \to j_!j^*F \to F \to i_*i^*F \to 0 $$
Now if the variety $X$ is not proper, you can always find a dense open immersion $u:X\hookrightarrow \overline{X}$ into a proper variety. Since $u_!$ is exact, that $u_!i_* = u_!i_! = (ui)_!$ and applying $H^*(\overline{X},-)$ we obtain a long exact sequence $$ \to H^{i}_c(U,j^*F) \to H^{i}_c(X,F) \to H^{i}_c(Z,i^*F) \to H^{i+1}_c(U,j^*F) \to $$
For the computation $H^*_c(\mathbb{A}^n)$ it reduces to that of $H^*_c(\mathbb{P}^n)$ by the localizing sequence. But since $\mathbb{P}^n$ is proper, we have $H^{*}_c(\mathbb{P}^n) = H^*(\mathbb{P}^n) = K[h]/(h^{n+1})$ where $h$ is the class of any hyperplane and has degree 2. Moreover, the morphism $\mathbb{P}^n \hookrightarrow \mathbb{P}^{n+1}$ induces the natural projection.