UNIFORMLY RIGID SPACES 1. Introduction Let K be a non ...

40 CHRISTIAN KAPPENA = R[[S]], then M(A) is the closure of the Berkovich open unit disc in theBerkovich closed unit disc, which is obtained by adding the Gauss point.The corresponding element of M(R[[S]] ⊗ R K) is precisely the Gauss normdiscussed above.The formation of M(A) does not behave well with respect to localization:If A = R〈X, Y 〉[[Z]]/(XY − Z), equipped with the Jacobson-adic topology,and if B = A {X−Y } , then the induced map M(B) → M(A) is not injective,because the formally smooth R-algebra A has no nontrivial idempotents,while B is a direct sum of two domains, such that M(B) contains twoGauss points mapping to a single Gauss point in M(A). In particular, theformation of M(A) does not globalize. Nonetheless, we think that a quasicompactuniformly rigid K-space X should be viewed as a compactificationof its underlying rigid K-space X r . This should be made more precise bystudying the topos of X.References[1] Vladimir Berkovich. Spectral theory and analytic geometry over non-archimedeanfields, volume 33 of Mathematical Surveys and Monographs. American MathematicalSociety, 1990.[2] Vladimir Berkovich. Vanishing cycles for formal schemes II. Invent. Math.,125(2):367–390, 1996.[3] Pierre Berthelot. Cohomologie rigide et cohomologie rigide à supports propres.Prépublication de l’université de Rennes 1, 1996.[4] Siegfried Bosch. Lectures on formal and rigid geometry. Preprint series of the SFBGeometrische Strukturen in der Mathematik, Münster, 378, 2005.[5] Siegfried Bosch, Ulrich Güntzer, and Reinhold Remmert. Non-Archimedean analysis.A systematic approach to rigid analytic geometry, volume 261 of Grundlehrender Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1984.[6] Siegfried Bosch and Werner Lütkebohmert. Formal and rigid geometry I. Rigidspaces. Math. Ann., 295:291–317, 1993.[7] Siegfried Bosch and Werner Lütkebohmert. Formal and rigid geometry II. Flatteningtechniques. Math. Ann., 296(3):403–429, 1993.[8] Nicolas Bourbaki. Commutative algebra, volume Chapters 1–7 of Elements of Mathematics.Springer-Verlag, 1998.[9] Ching-Li Chai. A bisection of the artin conductor. unpublished, http://www.math.upenn.edu/~chai/papers_pdf/bAcond_v21.pdf.[10] Brian Conrad. Irreducible components of rigid spaces. Ann. Inst. Fourier (Grenoble),49(2):473–541, 1999.[11] Johan de Jong. Crystalline Dieudonné module theory via formal and rigid geometry.Inst. Hautes Études Sci. Publ. Math., 82:5–96, 1995.[12] Johan de Jong. Erratum to: ”Crystalline Dieudonné module theory via formal andrigid geometry”. Inst. Hautes Études Sci. Publ. Math., 87:175–175, 1998.[13] David Eisenbud. Commutative algebra with a view towards algebraic geometry, volume150 of Graduate Texts in Mathematics. Springer-Verlag, 1995.[14] Alexander Grothendieck and Jean Dieudonné. Éléments de géométrie algébriqueIII. étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ.Math., 11:167 pp., 1961.