We have a variety $X$, and we can make a site $X_{et}$, which then gives us an abeliean category of sheaves from $X$ to abelian groups. Given a continuous map $f:X\to Y$, get functors $f^$ and $f_$, which are adjoints, i.e.:

\[\hom_{\shv(X)}(f^*A, B) \cong \hom_{\shv(Y)}(A, f_*B)\]

This can be used to define the functor $f^ * $. The isomorphism is given by $u:A\to f_ * B$ by $u(U):A(U)\to f _ * (B)(U)\coloneq B(f^{-1}(U))$. Given $V$ an etal morphism to $X$ and $U$ an etal morphism to $Y$ such that $f(V)\subseteq U$ (meaning that $f(V)$ is an etal morphism to $U$), we get $A(U)\xrightarrow{u(U)}B(f^{-1}(U))\xrightarrow{\res}B(V)$, and this diagram is compatable with maps between $U$’s and $V$’s. Take the colimit, and get $f^{ * \text{pre}}\lim_{U:f(V)\subset U}A(U)\to B(V)$. Using the adjunction between the sheafification functor and the forgetful functor, we obtain that this induces a map between sheaves.

A special case of this is the inclusion of an open and closed subvariety. Suppose $i:Z\hookrightarrow X$ is a closed immersion, and $j:U\subset X$ is an open immersion. (Note: This style of thinking is apperently very common; read: birational geometry and then blowups). Given a sheaf $\mathscr F$ on $X$, we have $i^* \mathscr F$ and $j^* \mathscr F$. Then we can consider $j^* \mathscr f\xrightarrow{id} j^* \mathscr F$, which gives by adjunction $\mathscr F \xrightarrow{unit} j_* j^* \mathscr F$. Then we have $i^* \mathscr F\to i^* j_* j^* \mathscr F$. Call $i^* \mathscr F$ the restriction $\mathscr F_Z$, and $j^* \mathscr F$ the restriction $\mathscr F_U$. Then we ave te gluing map $\mathscr F_Z\to i^* j_* \mathscr F_U$. It is a proposition that there is an equivalence of categories $\shv(X_{\et})\cong {(\mathscr F_Z, \mathscr F_U, \mathscr F_Z\to i^* j_* \mathscr F_U)}$. Then $i^* =\mathscr F_Z$ and we have a map $\mathscr F\to i_* \mathscr F_Z$. Get the diagram

\[\mathscr F \to i_*\mathscr F_Z\]
\[i_*\mathscr F_U \xrightarrow{unit} i_*i^*j_* \mathscr F\]

with vertical arrows. Furthermore, this diagram is a fiber product / pullback.

Remark: The $j^ * $ has a left adjoint. Def. (The lower shreak functor / extension by zero).

Remark: $i^* j_! \mathscr F = 0$.

Remark: The functor $i_* $ has a right adjoint $i^!$.

Def. Given $\mathscr F\in \sh(X_{\et})$, get

This is $i^!$.

Remark: Then, we have

- $i^!$ is a right adjoint to $i_*$ is a right adjoint to $i^*$

Remark: We can do a lot of this for sheaves of sets, with the exception of the kernel stuff, and we need to alter the extension by zero to be an extension by a pointed set.

Def. Relative Cohomology / Cohomology with Supports. Keep the same notation. Define a functor $\Gamma_Z:\sh(X_{\et})\to Ab$, by $\Gamma_Z(X, \mathscr F) = \ker(\Gamma(X, \mathscr F) \to \Gamma(U, \mathscr F))$. This is all the sections of $\mathscr F$ which vanish identically on $U$, which we can rephrase as “Sections of $\mathscr F$ supported on $\mathscr Z$”.

Ex. If $Z\subsetneq X$, then $\Gamma_Z(X, \underline{\mathbb Z}) = 0$.

We then define cohomolgy with $Z$-supports / cohomology relative to $Z$ to be $R^r\Gamma_Z(X, \mathscr F)$.

Then next question to ask is how does this relate to cohomology of $X$ and $U$? The answer is a long exact sequence:

This comes from

because we have $H^r(X, j_* j^* \mathscr F) = H^r(U, j^* \mathscr F)$, because $j^* $ is exact (it has left and right adjoints) and then an argument with leray spectral sequences et. cetera gives the intended equality.

Prop. For $U$ a nonsingular curve over $\bar k = k$ for $x\in |U|$ a closed point, if $\char k$ does not divide $n$, then

\[H^r_{\{x\}}(U, \mu_n)\cong \begin{cases}
    \mathbb Z/ n & r=2\\
    0 & r\neq 2\\
\end{cases}\]

Proof. (Sketch) The cohomology $H^r_{{x}}(U, \mu_n)$ depends only on a neighborhood of $x$. Further, this implies that $H^r_{{x}}\cong H^r_{{v}})(V, \mu_n)$ where $V\to U$ is etal, and $v\mapsto u$. We can do this for all $V\to U$, and so we take the limit

\[\lim_{(V, v)\to (U, u)}\mc O(V)\]

If this were the Zariski topology, you’d get the local ring at $x$; however, here, you get the hensilization of the local ring $\mc O_{U, x}$. What is hensilization?

Ex. Suppose that $U$ is the affine line $\spec k[t]$. Taking the local ring at the origin is $k[t][1/f]_ {f(0)\neq 0}$, a DVR with maximal ideal $(t)$ and residue field $k$. Then the hensilization $\hat\mc O_{U, {0}}\subset \overline{k(t)}$ contains all roots of polynomials that split mod $t$. The reason for this is because polynomials which split mod $t$ are polynomials that give etal maps to a neighborhood of 0 (?).

Then we obtain $H^r_{{x}}(U, \mu_n) = H^r_{{x}}(\spec \mc O_{U, x}^h, \mu)$. Then from our discussion a few weeks ago, we have that the cohomology of $V$ with coefficients in $\mathbb G_{m}$ is $V^* $ when $r\neq 0$, $\pic V = 0$ when $r=1$ (because the picaird group of a spectrum of a DVR is trivial) and finally 0 in higher degree by Tsen’s theorem. Then we have that the higher cohomology of $V\setminus {x}$ is zero as well, by a similar argument. (Recall that $V-{x} =\spec k$). Then we have

And the final term is $\mathbb Z$. Now we use the Kummer sequence.

The middle two terms are integers, the first term is zero, so the last term is $\mathbb Z/n\mathbb Z$. A restatement of all of this in terms of a curve $U$ with an inclusion $i:{x}\to U$ is as follows. We can consider the derived shreak of $i$; $R^ri^!\mu_n = \mathbb Z/n\mathbb Z$ is the same as $H^r(U, \mu_n)$.

Remark: This is a special case of “cohomological purity” (the local version):

Suppose we have smooth embedded varieties $i:Z\to X$, and say the codimension of this embedding is $C$. Let $\mathscr F$ be a locally constant sheaf of $\mathbb Z/ n\mathbb Z$- modules (for example, $\underline{\mathbb Z/ n\mathbb Z}$. Then

where the operation $\bullet(-c)$ is $\mathscr F\otimes \mu^{\otimes (-c)}$.