In Demazure's Lectures on p-Divisible Groups, an affine formal scheme over a field $k$ is defined as $\mathop{\mathrm{Spf}}(A)$ for some profinite $k$-algebra $A$. However, in EGA I Chap. 10, an affine formal scheme is defined as $\mathop{\mathrm{Spf}}(A)$ for some admissible ring $A$. My question is: Is Demazure's definition a special case of EGA's definition? If not, is there a more general definition which contains both of them as special cases.

  • $\begingroup$ Your question is not about formal shemes, but about if a profinite $k$-algebra, for $k$ a field, is an admissible (topological) ring (with the topology given by being profinite). And the answer is clearly affirmative. So Demazure's definition can be seen as a special case of EGA's definition. $\endgroup$ Jun 13, 2020 at 8:52
  • $\begingroup$ @Nulhomologous But it seems that an infinite product of $k$ is a profinite $k$-algebra which is not admissible. $\endgroup$ Jun 13, 2020 at 9:29
  • $\begingroup$ I don't see how an infinite product of k could be a profinite k-algebra. $\endgroup$ Jun 13, 2020 at 19:42
  • $\begingroup$ @Nulhomologous According to Demazure, a profinite $k$-algebra is equivalent to a left exact functor from the cat $\mathcal C$ of finite $k$-algebras to the cat of sets. Let $I$ be any set. Then the constant sheaf on $\mathcal C$ with value in $I$ (i.e. $R \mapsto$ locally constant function from $\mathop{\mathrm{Spec}}(R)$ to $I$) corresponds to the product of copies $k_i$ of $k$ indexed by $I$. In particular, the constant $p$-divisible group $\mathbb Q_p/\mathbb Z_p$ over $k$ should corresponds to an infinite product of $k$. $\endgroup$ Jun 14, 2020 at 1:49


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