Approximation of versal deformations.
Let \(S\) be an excellent scheme and \(F\) a category cofibred in groupoids over the category of \(S\)-schemes. Roughly means that \(F(T)\) is the category of geometric structures over a \(S\)-scheme \(T\), which behave well with respect to the base change. \(F\) is locally of finite presentation over \(S\) if for any directed system \((A_i)\) of \({\mathcal O}_S\)-algebras with direct limit \(A\), the natural transformation of categories from the inductive limit of \(F(A_i)\) to \(F(A)\) is fully faithful and essential surjective. Let \({\bar F}(T)\) be the set of isomorphism classes of objects in \(F(T)\) and suppose that \(F\) satisfies the so called Schlessinger-Rim criteria and the natural map from \({\bar F}(B)\) to the projective limit of \({\bar F}(B/m_B^{n+1})\) has dense image for all complete local noetherian \({\mathcal O}_S\)-algebras \((B,m_B)\) with \(B/m_B\) residually finite over \(S\). For any residually finite \({\mathcal O}_S\)-field \(k\) and any object \(\rho\) in \(F(k)\), any formal versal deformation of \(\rho\) in \(F(k)\) is algebraizable. A similar result was claimed by the reviewer and \textit{M. Roczen} in [Rev. Roum. Math. Pures Appl. 33, 251--260 (1988; Zbl 0661.14003)] but the proof has a gap.
scientific article; zbMATH DE number 1868016
scientific article