In Lei Fu's "Etale Cohomology Theory", theorem 8.6.7 is Lefschetz's Trace formula for curves, it is stated as follow:
Theorem: Let $X$ be a smooth projective curve over an algebraically closed field $k$, $n$ an integer invertible in $X$, $h: X \to X$ a $k$-morphism, $\Gamma_h: X \to X \times_k X$ the graph of $h$, $\Delta$ the divisor of $X \times_k X$ defined by the diagonal morphism $\Delta: X \to X \times_kX$, and $\Gamma^*_h(\Delta)$ the pulling back of the divisor $\Delta$. Then we have \begin{align} \sum\limits_{q=0}^2 (-1)^q\mathrm{Tr}(h^*, H^q_{et}(X, \mathbb{Z}/n\mathbb{Z})) \equiv \mathrm{deg}(\Gamma^*_h(\Delta))\mod n \end{align}
I have trouble interpreting what $\Gamma^*_h(\Delta)$ really is. I understand that it is meant to represent the intersection of the diagonal and the graph of $h$ in $X \times_k X$, projected in $X$, to represent the fixed points of $h$. In particular, if $h$ has as only finitely many isolated fixed points $x_1,\ldots, x_r$ with multiplicity $m_1,\cdots,m_r$, then $\Gamma^*_h(\Delta)$ should be the divisor on $X$ defined as $\sum_k m_k[x_k]$.
But I know that divisors don't pull back that easily, for instance is $h$ is simply the identity, I don't know what divisor $\Gamma_h^*(\Delta)$ on X could be.
As I interpret it, $\Gamma^*_h$ is not in general a well defined divisor, but rather the image by $\Gamma^*_h : H^{2}(X \times_k X, \mathbb{Z}/n\mathbb{Z}) \to H^{2}(X, \mathbb{Z}/n\mathbb{Z})$ of the class of the divisor $\Delta$ seen in $Pic(X)/nPic(X) \cong H^2(X, \mathbb{Z}/n\mathbb{Z})$ through the connection morphism $H^1(X, \mathcal{O}^*_{X_{et}}) \to H^2(X, \mathbb{Z}/n\mathbb{Z})$ given by Kummer's theory and the short exact sequence \begin{align} 0 \longrightarrow \mathbb{Z} \longrightarrow \mathcal{O}^*_{X_{et}} \longrightarrow \mathcal{O}^*_{X_{et}} \longrightarrow 0 \end{align} Then $\deg(\Gamma_h^*(\Delta)) \mod n$ makes sense as $\overline{deg}(\underbrace{\Gamma_h^*(\Delta)}_{\in H^2(X, \mathbb{Z}/n\mathbb{Z})})$ where $\overline{deg} : \mathrm{Pic}(X)/n\mathrm{Pic}(X) \to \mathbb{Z}/n\mathbb{Z}$ arises from $\mathrm{deg} : \mathrm{Pic} \to \mathbb{Z}$.
This interpretation does not seems to change anything in the proof given by the author, except when he writes: \begin{align} \mathrm{cl}(\Delta) = \mathrm{cl}(\mathcal{L}(\Delta)) = \mathrm{cl(\mathcal{L}(\Gamma^*_h(\Delta)))} \end{align}
Where $\mathrm{cl}$ is the class in $H^2(X \times_k X, \mathbb{Z}/n)$ or $H^2(X, \mathbb{Z}/n\mathbb{Z})$ (depending of the argument) of a cycle, and $\mathcal{L}(D)$ is an invertible $\mathcal{O}_X$ module associated to the divisor D. It is shown that in this case the image of its class in $H^2(X, \mathbb{Z}/n\mathbb{Z})$ is the same as $\mathrm{cl}(D)$. The first equality remains true but the second makes no sense if $\Gamma_h^*(\Delta)$ is not a divisor. But since the trace of such an expression is then taken, and that the Trace $H^2(X, \mathbb{Z}/n\mathbb{Z}) \to \mathbb{Z}/n\mathbb{Z}$, is defined as the composite $H^2(X, \mathbb{Z}/n\mathbb{Z}) \cong \mathrm{Pic}(X)/n\mathrm{Pic}(X) \overset{\overline{\mathrm{deg}}}{\to} \mathbb{Z}/n\mathbb{Z}$, this does not seem to impact the proof.
I still have two questions
- Is my interpretation correct? I think that as I said, this interpretation of $\Gamma_h^*(\Delta)$ does not change anything to the proof, but I am still a bit unsure.
- If it is correct, how does this interpretation translates back to the "nice" interpretation, in the case of a function with isolated fixed points $x_1,\cdots, x_r$ with multiplicities $m_1,\cdots,m_r$, so that $\mathrm{deg}(\Gamma_h^*(\Delta)) = \sum_k m_k$ as expected?