# Torelli Over an Algebraically Closed Field

I was originally going to do this post in wordpress but then instead I’m just posting Torelli over k bar; a rough outline based on a proof in Polishchuk’s book.

But I wanted to add some notes here.  At some point I reference this post about chapter 17 in Polishchuk

Extensively in the proof base change is used.  This says if there is a diagram

$\begin{array}{ccc} A & \xrightarrow{f} & B\\ \downarrow q & \empty & \downarrow p\\ C & \xrightarrow{g} & D \end{array}$

where $A = B\times_D C$ and $g$ if flat and $p$ is proper then there is an isomorphism

$Rq_*f^*H \cong g^*Rp_*H$

as $f,g$ are flat they don’t need to be derived (Huybrechts pg 85).  As an application consider three schemes $A',A,B$ and a map $g \colon A' \to A$.  Note $A\times B \to B$ is flat.  Let $F \in D^b(A'),\ P \in D^b(A \times B)$ and consider $\Phi_P(Rg_*F) = Rp_{2*}(p_1^*g_*F\otimes P)$.

$\begin{array}{ccc} A' \times B & \xrightarrow{q_1} & A'\\ \downarrow g\times id & \empty & \downarrow g\\ A \times B & \xrightarrow{p_1} & A \end{array}$

This gives $R(g\times id)_*q_1^*F \cong p_1^*Rg_*F$.  Also $q_2 = p_2 \circ (g \times id)$. So

$\Phi_{(g \times id)^*P}(F) = Rq_{2*}(q_1^*F\otimes (g\times id)^*P)$

$= Rp_{2*}R(g \times id)_*(q_1^*F \otimes (g\times id)^*P)$

$= Rp_{2*}(R(g\times id)_*q_1^*F \otimes P)$ (proj. form.)

$Rp_{2*}(p_1^*Rg_*F \otimes P) = \Phi_P(Rg_*F)$

This is the result for

$A' \to A \leftarrow A \times B \to B$

There are similar stories for

$A' \leftarrow A \leftarrow A \times B \to B$

$A \leftarrow A \times B \to B \to B'$

$A \leftarrow A \times B \to B \leftarrow B'$