108.68 An algebraic stack not satisfying strong formal effectiveness

This is [Example 4.12, Bhatt-Algebraize]. Let $k$ be an algebraically closed field. Let $A$ be an abelian variety over $k$. Assume that $A(k)$ is not torsion (this always holds if $k$ is not the algebraic closure of a finite field). Let $\mathcal{X} = [\mathop{\mathrm{Spec}}(k)/A]$. We claim there exists an ideal $I \subset k[x, y]$ such that

$\mathcal{X}_{\mathop{\mathrm{Spec}}(k[x, y]^\wedge )} \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{X}_{\mathop{\mathrm{Spec}}(k[x, y]/I^ n)}$

is not essentially surjective. Namely, let $I$ be the ideal generated by $xy(x + y - 1)$. Then $X_0 = V(I)$ consists of three copies of $\mathbf{A}^1_ k$ glued into a triangle at three points. Hence we can make an infinite order torsor $P_0$ for $A$ over $X_0$ by taking the trivial torsor over the irreducible components of $X_0$ and glueing using translation by nontorsion points. Exactly as in the proof of Lemma 108.67.3 we can lift $P_0$ to a torsor $P_ n$ over $X_ n = \mathop{\mathrm{Spec}}(k[x, y]/I^ n)$. Since $k[x, y]^\wedge$ is a two dimensional regular domain we see that any torsor $P$ for $A$ over $\mathop{\mathrm{Spec}}(k[x, y]^\wedge )$ is torsion (Lemmas 108.67.1 and 108.67.2). Hence the system of torsors is not in the image of the displayed functor.

Lemma 108.68.1. Let $k$ be an algebraically closed field which is not the closure of a finite field. Let $A$ be an abelian variety over $k$. Let $\mathcal{X} = [\mathop{\mathrm{Spec}}(k)/A]$. There exists an inverse system of $k$-algebras $R_ n$ with surjective transition maps whose kernels are locally nilpotent and a system $(\xi _ n)$ of $\mathcal{X}$ lying over the system $(\mathop{\mathrm{Spec}}(R_ n))$ such that this system is not effective in the sense of Artin's Axioms, Remark 96.20.2.

Proof. See discussion above. $\square$

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