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Étude explicite de quelques n-champs géométriques

Abstract : In [PRID], Pridham has shown that any Artin n-stack M has a presentation as a simplicial scheme X. → M such that the simplicial scheme X satisfies certain properties denoted G.Pn,k of [GROTH]. In the presentation (…→ X2 → X1 → X0 → M), the scheme X1 represents a chart for X0 x MX0. Thus, the smoothness of X0 → M is equivalent to the smoothness of the two projections ә0,ә1 : X1 → X0. These are the first two parts of the Grothendieck-Pridham condition, denoted G.P1,0 and G.P1,1. In [BENZ12] we introduced an Artin n-stack M of Maurer-Cartan elements of a dg-category. We constructed a chart, and have already proven the first smoothness conditions explicitly. For any n and any 0 ≤ k ≤ n Pridham considers a scheme denoted MatchΛkn(X) with a morphism Xn → MatchΛkn(X). We will construct explicitly the Grothendieck-Pridham simplicial scheme and show the smoothness of the preceding map, therefore M is a geometric n-stack.
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Submitted on : Wednesday, October 2, 2013 - 1:13:20 AM
Last modification on : Thursday, August 4, 2022 - 4:58:07 PM
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Brahim Benzeghli. Étude explicite de quelques n-champs géométriques. Mathématiques générales [math.GM]. Université Nice Sophia Antipolis, 2013. Français. ⟨NNT : 2013NICE4032⟩. ⟨tel-00868795⟩



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