The Stacks project

Example 89.9.6. Here is a more explicit example of an $R$ as in Lemma 89.9.5. Let $p$ be a prime number and let $n \in \mathbf{N}$. Let $\Lambda = \mathbf{F}_ p(t_1, t_2, \ldots , t_ n)$ and let $k = \mathbf{F}_ p(x_1, \ldots , x_ n)$ with map $\Lambda \to k$ given by $t_ i \mapsto x_ i^ p$. Then we can take

\[ R = \Lambda [x_1, \ldots , x_ n]^\wedge _{(x_1^ p - t_1, \ldots , x_ n^ p - t_ n)} \]

We cannot do “better” in this example, i.e., we cannot approximate $\mathcal{C}_\Lambda $ by a smaller smooth object of $\widehat{\mathcal{C}}_\Lambda $ (one can argue that the dimension of $R$ has to be at least $n$ since the map $\Omega _{R/\Lambda } \otimes _ R k \to \Omega _{k/\Lambda }$ is surjective). We will discuss this phenomenon later in more detail.

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