The Stacks project

Example 98.13.2. Let $k$ be a field and set $\Lambda = k[s, t]$. Consider the functor $F : \Lambda \text{-algebras} \longrightarrow \textit{Sets}$ defined by the rule

\[ F(A) = \left\{ \begin{matrix} * & \text{if there exist }f_1, \ldots , f_ n \in A\text{ such that } \\ & A = (s, t, f_1, \ldots , f_ n)\text{ and } f_ i s = 0\ \forall i \\ \emptyset & \text{else} \end{matrix} \right. \]

Geometrically $F(A) = *$ means there exists a quasi-compact open neighbourhood $W$ of $V(s, t) \subset \mathop{\mathrm{Spec}}(A)$ such that $s|_ W = 0$. Let $\mathcal{X} \subset (\mathit{Sch}/\mathop{\mathrm{Spec}}(\Lambda ))_{fppf}$ be the full subcategory consisting of schemes $T$ which have an affine open covering $T = \bigcup \mathop{\mathrm{Spec}}(A_ j)$ with $F(A_ j) = *$ for all $j$. Then $\mathcal{X}$ satisfies [0], [1], [2], [3], and [4] but not [5]. Namely, over $U = \mathop{\mathrm{Spec}}(k[s, t]/(s))$ there exists an object $x$ which is versal at $u_0 = (s, t)$ but not at any other point. Details omitted.

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View Example 98.13.2 as pdf