The Stacks project

108.66 The stack of proper algebraic spaces is not algebraic

In Quot, Section 97.13 we introduced and studied the stack in groupoids

\[ p'_{fp, flat, proper} : \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \longrightarrow \mathit{Sch}_{fppf} \]

the stack whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$. We proved that this satisfies many of Artin's axioms. In this section we why this stack is not algebraic by showing that formal effectiveness fails in general.

The canonical example uses that the universal deformation space of an abelian variety of dimension $g$ has $g^2$ formal parameters whereas any effective formal deformation can be defined over a complete local ring of dimension $\leq g(g + 1)/2$. Our example will be constructed by writing down a suitable non-effective deformation of a K3 surface. We will only sketch the argument and not give all the details.

Let $k = \mathbf{C}$ be the field of complex numbers. Let $X \subset \mathbf{P}^3_ k$ be a smooth degree $4$ surface over $k$. We have $\omega _ X \cong \Omega ^2_{X/k} \cong \mathcal{O}_ X$. Finally, we have $\dim _ k H^0(X, T_{X/k}) = 0$, $\dim _ k H^1(X, T_{X/k}) = 20$, and $\dim _ k H^2(X, T_{X/k}) = 0$. Since $L_{X/k} = \Omega _{X/k}$ because $X$ is smooth over $k$, and since $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X) = H^ i(X, T_{X/k})$, and because we have Cotangent, Lemma 90.23.1 we find that there is a universal deformation of $X$ over

\[ k[[x_1, \ldots , x_{20}]] \]

Suppose that this universal deformation is effective (as in Artin's Axioms, Section 96.9). Then we would get a flat, proper morphism

\[ f : Y \longrightarrow \mathop{\mathrm{Spec}}(k[[x_1, \ldots , x_{20}]]) \]

where $Y$ is an algebraic space recovering the universal deformation. This is impossible for the following reason. Since $Y$ is separated we can find an affine open subscheme $V \subset Y$. Since the special fibre $X$ of $Y$ is smooth, we see that $f$ is smooth. Hence $Y$ is regular being smooth over regular and it follows that the complement $D$ of $V$ in $Y$ is an effective Cartier divisor. Then $\mathcal{O}_ Y(D)$ is a nontrivial element of $\mathop{\mathrm{Pic}}\nolimits (Y)$ (to prove this you show that the complement of a nonempty affine open in a proper smooth algebraic space over a field is always a nontrivial in the Picard group and you apply this to the generic fibre of $f$). Finally, to get a contradiction, we show that $\mathop{\mathrm{Pic}}\nolimits (Y) = 0$. Namely, the map $\mathop{\mathrm{Pic}}\nolimits (Y) \to \mathop{\mathrm{Pic}}\nolimits (X)$ is injective, because $H^1(X, \mathcal{O}_ X) = 0$ (hence all deformations of $\mathcal{O}_ X$ to $Y \times \mathop{\mathrm{Spec}}(k[[x_ i]]/\mathfrak m^ n)$ are trivial) and Grothendieck's existence theorem (which says that coherent modules giving rise to the same sheaves on thickenings are isomorphic). If $X$ is general enough, then $\mathop{\mathrm{Pic}}\nolimits (X) = \mathbf{Z}$ generated by $\mathcal{O}_ X(1)$. Hence it suffices to show that $\mathcal{O}_ X(n)$, $n > 0$ does not deform to the first order neighbourhood1. Consider the cup-product

\[ H^1(X, \Omega _{X/k}) \times H^1(X, T_{X/k}) \longrightarrow H^2(X, \mathcal{O}_ X) \]

This is a nondegenerate pairing by coherent duality. A computation shows that the Chern class $c_1(\mathcal{O}_ X(n)) \in H^1(X, \Omega _{X/k})$ in Hodge cohomology is nonzero. Hence there is a first order deformation whose cup product with $c_1(\mathcal{O}_ X(n))$ is nonzero. Then finally, one shows this cup product is the obstruction class to lifting.

Lemma 108.66.1. The stack in groupoids

\[ p'_{fp, flat, proper} : \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \longrightarrow \mathit{Sch}_{fppf} \]

whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$ (see Quot, Section 97.13) is not an algebraic stack.

Proof. If it was an algebraic stack, then every formal object would be effective, see Artin's Axioms, Lemma 96.9.5. The discussion above show this is not the case after base change to $\mathop{\mathrm{Spec}}(\mathbf{C})$. Hence the conclusion. $\square$

[1] This argument works as long as the map $c_1 : \mathop{\mathrm{Pic}}\nolimits (X) \to H^1(X, \Omega _{X/k})$ is injective, which is true for $k$ any field of characteristic zero and any smooth hypersurface $X$ of degree $4$ in $\mathbf{P}^3_ k$.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0D1Q. Beware of the difference between the letter 'O' and the digit '0'.