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Our aim is to explain how to use Olsson’s theorem on proper hypercoverings of Artin stacks (the main result in [4]) to prove the formal GAGA theorem and the Grothendieck existence theorem for proper Artin stacks over an adic noetherian ring. In [4] a proof of this result is given from a different point of view. Everything in the first two sections below is definitions and “general nonsense”. 1. Completion of coherent sheaves and stacks Let X be a locally noetherian stack and X0 ⊆ |X | a closed subset. Definition 1.1. For F ∈ Coh(X), the completion of F along X0 is the sheaf ̂F = lim F /JαF ←−α on Xlis-ét, where {Jα} is the inverse system of coherent ideals on X with zero locus X0. Define the sheaf of rings O cX def = ÔX = li

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this site. By [3, 12.7.4, the category of cartesian OcX-modules on Xlis-ét is equivalent to the category of OcXét-modules on X, 1.1) ModXlis-ét, cart(O cX) ≃ ModXét (O c Xét

Year: 2011

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oai:CiteSeerX.psu:10.1.1.191.181

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