From ba02c670badc6903f9f230c4e530f4623bc666cf Mon Sep 17 00:00:00 2001 From: Tim Hosgood Date: Wed, 20 Mar 2024 14:50:04 +0100 Subject: [PATCH] finished i.8 --- trees/sga1/sga1-i/sga1-i.8.tree | 39 +++++++++++++++++++++++++++++++-- 1 file changed, 37 insertions(+), 2 deletions(-) diff --git a/trees/sga1/sga1-i/sga1-i.8.tree b/trees/sga1/sga1-i/sga1-i.8.tree index 90f5f3b..76bc0bf 100644 --- a/trees/sga1/sga1-i/sga1-i.8.tree +++ b/trees/sga1/sga1-i/sga1-i.8.tree @@ -68,6 +68,41 @@ } \proof{ - By + By \ref{sga1-i.5.5}, it remains only to show that every étale #{S_0}-scheme #{X_0} is isomorphic to an #{S_0}-scheme #{X\times_S S_0}, where #{X} is an étale #{S}-scheme. + The underlying topological space of #{X} must necessarily be identical to the one of #{X_0}, and with #{X_0} being identified with a closed sub-prescheme of #{X}. + The problem is thus equivalent to the following: + find, on the underlying topological space #{|X_0|} of #{X_0}, a sheaf of algebras #{\sh{O}_X} over #{f_0^*(\sh{O}_S)} (where #{f_0} is the projection #{X_0\to S_0}, thought of here as a continuous map of the underlying spaces) that makes #{|X_0|} an étale #{S}-prescheme #{X}, as well as an algebra homomorphism #{\sh{O}_X\to\sh{O}_{X_0}} that is compatible with the homomorphism #{f_0^*(\sh{O}_S)\to f_0^*(\sh{O}_{S_0})} on the sheaves of scalars, and that induces an isomorphism #{\sh{O}_X\otimes_{f_0^*(\sh{O}_S)*}f_0^*(\sh{O}_{S_0})\xrightarrow{\sim}\sh{O}_{X_0}}. + (Then #{X} will be an étale #{S}-prescheme that is reduced along #{X_0}, and thus separated over #{S}, since #{X_0} is separated over #{S_0}, and #{X} satisfies all the desired properties). + + If #{(U_i)} is an open cover of #{X_0}, and if we find a solution to the problem on each of the #{U_i}, then it follows from the uniqueness theorem \ref{sga1-i.5.5} that these solutions glue (i.e. the sheaves of algebras that they define, endowed with their augmentation homomorphisms, glue), and we claim that the ringed space thus constructed over #{S} is an étale #{S}-prescheme #{X} endowed with an isomorphism #{X\times_S S_0\xleftarrow{\sim}X_0}. + It thus suffices to find a solution locally, which we know is possible by \ref{sga1-i.8.1}. + } +} + +\subtree[sga1-i.8.4]{ + \taxon{corollary} + \number{I.8.4} + + \p{ + Let #{S} be a locally Noetherian formal prescheme, endowed with an ideal of definition #{\sh{J}}, and let #{S_0=(|S|,\sh{O}_S/\sh{J})} be the corresponding ordinary prescheme. + Then the functor #{\mathfrak{X}\mapsto\mathfrak{X}\times_S S_0} from the category of étale covers of #{S} to the category of étale covers of #{S_0} is an equivalence of categories. + } + + \proof{ + \p{ + Of course, we define an étale cover of a \em{formal} prescheme #{S} to be a cover of #{S} (i.e a formal prescheme over #{S} defined by a coherent sheaf of algebras #{\sh{B}}) such that #{\sh{B}} is \em{locally free}, and such that the residue fibres #{\sh{B}_s\otimes_{\sh{O}_s}k(s)} of #{\sh{B}} are \em{separable} algebras over #{k(s)}. + If we denote by #{S_n} the ordinary prescheme #{(|S|,\sh{O}_S/\sh{J}^{n+1})}, then the data of a coherent sheaf of algebras #{\sh{B}} on #{S} is equivalent to the data of a sequence of coherent sheaves of algebras #{\sh{B}_n} on the #{S_n}, endowed with a transitive system of homomorphisms #{\sh{B}_m\to\sh{B}_n} (for #{m\geq n}) defining the isomorphisms #{\sh{B}_m\otimes_{\sh{O}_{S_m}}\sh{O}_{S_n}\xrightarrow{\sim}\sh{B}_n}. + It is immediate that #{\sh{B}} is locally free if and only if the #{\sh{B}_n} are locally free over the #{S_n}, and that the separability condition is satisfied if and only if it is satisfied for #{\sh{B}_0}, or for all the #{\sh{B}_n}. + Thus #{\sh{B}} is étale over #{S} if and only if the #{\sh{B}_n} are étale over the #{S_n}. + Taking this into account, \ref{sga1-i.8.4} follows immediately from \ref{sga1-i.8.3}. + } } -} \ No newline at end of file +} + +\subtree{ + \taxon{remark} + + \p{ + It was not necessary to restrict ourselves to the case of \em{covers} in \ref{sga1-i.8.4}, but this is the only case that we will use for the moment. + } +}