Example 90.9.6. Here is a more explicit example of an R as in Lemma 90.9.5. 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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