Let $A$ be an abelian variety over a base scheme $S$, write $\pi: A \rightarrow S$, equipped with the zero section $e: S \rightarrow A$. Let $A^{\vee}$ be the dual variety of $A$. I am hoping to obtain some kind of result like $$ \underline{\mathrm{Lie}}(A^{\vee}/S) \cong \mathrm{R}^1 \pi_{\ast} \mathcal{O}_A. \quad (\dagger) $$ Here $\underline{\mathrm{Lie}}(A/S)$ is the Lie algebra sheaf of $A_{/S}$.
My questions are:
- I am currently having no idea of a precise definition of the Lie algebra sheaf. I found in Mumford's book Abelian Varieties but only saw the definition of the Lie algebra (page 94) associated to the group scheme $G_{/k}$ as a $k$-vector space. Where can I find a sheaf-version definition of the Lie algebra of abelian varieties that fits in the setup above?
- How to (find appropriate assumptions on $A/S$ and) show $(\dagger)$? Any proof, hint or references are welcome!
My attempt: I was told (though not knowing the precise defn of $\underline{\mathrm{Lie}}$) that there is an isomorphism for any group schemes $G$ over $S$: $$ \phi: \underline{\mathrm{Lie}}(G/S) \xrightarrow{\sim} \mathcal{Hom}_{\mathcal{O}_S}(\omega_{G/S}, \mathcal{O}_S) $$ as $\mathcal{O}_S$-modules, where $\omega_{G/S}$ is defined as the sheaf $e^{\ast} \mathcal{I}_{G/S}$, with $\mathcal{I}_{G/S}$ the sheaf of ideals of the closed immersion $e: S \rightarrow A$. Then I get stuck and have no idea how to carry on. :(
My motivation: I know for elliptic curves $E$ over a base scheme $S$ (at least in the world of char zero), we have a short exact sequence deduced from the Hodge-to-de Rham spectral sequence $$ 0 \rightarrow \pi_{\ast} \Omega^1_{E/S} \rightarrow \mathcal{H}^1_{\mathrm{dR}}(E/S) \rightarrow \mathrm{R}^1 \pi_{\ast} \mathcal{O}_E \rightarrow 0. $$ Meanwhile, I saw that people often write short exact sequences like $$ 0 \rightarrow \pi_{\ast} \Omega^1_{A/k} \rightarrow \mathcal{H}^1_{\mathrm{dR}}(A/k) \rightarrow \underline{\mathrm{Lie}}(A^{\vee}/k) \rightarrow 0. \quad (\star) $$ But I cannot find a proof of $(\star)$. Comparing the two, I am guessing some kind of result like $(\dagger)$, so that I can deduce $(\star)$ from "Hodge-to-de Rham".
Thank you so much for commenting and answering!
This is somewhat classical, but as it may be a little hard to find, let me make a comment here.
First, recall that one way to define $A^\vee$ is as the relative identity component of the Picard scheme: $A^\vee=\mathrm{Pic}^0_{A/S}$. The Lie algebra shea (which for a group $S$-scheme $G$ is defined as
$$\underline{\mathrm{Lie}}(G)(R)=\ker(G(R[\varepsilon])\to G(R))$$
where $\mathrm{Spec}(R)\to S$ is a morphism, and $R[\varepsilon]=R[x]/(x^2)$) doesn't detect the difference between the identity component and the full scheme. Thus, in your case the result you are after is Theorem 1, in Section 8.4 of Bosch--Lutkebohmert--Remmert's book Neron Models.