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We state and prove several kinds of analytification theorems of formal objects (such as coherent sheaves and formal complex spaces) which are in the spirit of Grothendieck's algebraization theorem in [EGA, III]. The formulation of the results was derived from deformation theory and especially M. Artin's work on representability of functors. The methods of proof depend heavily on a deeper study of cotangent complexes and resolvants. As applications one can deduce the convergence of formal versal deformations in diverse situations. A powerful tool for the solution of many moduli problems in algebraic geometry is Grothendieck's existence theorem in [EGA, III, Theoreme (5.1.4)]. This theorem gives a general algebraicity criterion for coherent formal sheaves and goes as follows. Theorem (Grothendieck). Let A be an adic noetherian ring, Y = Spec(A), > an ideal of def nition for A, Y' = V(>), f: X ) Y a separated morphism of finite type and X = f 1(Y ) . Furthermore, let Y = Y/yl = Spf(A), X = X/x, be the formal completions of Y with respect to Y' and of X with respect to X', A and f: X Y the completion of f . Then the functor S | ) S gives an equivalence between the category of coherent Ax-modules with proper support over Y and the category of coherent Mx-modules with proper support over Y . A generalization of this theorem to algebraic spaces (in the sense of Artin) can be found in [Knu, V, 6.3]. The applications in algebraic deformation theory are usually made there where one has to verify the efectiveness of formal deformations (compare for instance [Art, §1]). As the formulation of Grothendieck's theorem shows, a straightforward translation into the context of analytic geometry is not possible. But, having in mind the applications to deformation theory and especially the algebraization of formal moduli in the framework of analytic geometry (see [Bi2]), it is indeed possible to formulate an analogous assertion. The methods of Palamodov, developed in [Pal, Pa2] for the solution of the local moduli problem for compact complex spaces, and the expansion of these Received by the editors September 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 32G13, 14D15 (r) 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Grothendieck’s existence theorem in analytic geometry and related results