Huber, Roland A generalization of formal schemes and rigid analytic varieties. (English) Zbl 0814.14024 Math. Z. 217, No. 4, 513-551 (1994). There is a natural class of topological rings which contains both the affinoid rings (from rigid analytic geometry) and the noetherian adic rings. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring \(f\)-adic. For every \(f\)-adic ring \(A\) the topological space \(\text{Cont }A\) of all continuous valuations of \(A\) is a spectral space. If \(A\) has an open noetherian adic subring or if for every \(n \in N\) the ring \(A\langle X_ 1,\dots, X_ n\rangle\) of restricted power series in \(n\) variables over \(A\) is noetherian then there is a natural sheaf \({\mathcal O}_ A\) of topological rings on \(\text{Cont }A\). All stalks of \({\mathcal O}_ A\) are local rings. We call a topologically and locally ringed space which is locally isomorphic to some \(\text{Spa }A := (\text{Cont }A,{\mathcal O}_ A)\) adic. There are natural fully faithful functors from the category of rigid analytic varieties and the category of locally noetherian formal schemes to the category of adic spaces. In the first case one assigns to an affinoid rigid analytic variety \(\text{Sp }A\) the adic space \(\text{Spa }A\) and in the latter case one assigns to an affine noetherian formal scheme \(\text{Spf }A\) the adic space \(\text{Spa }A\). Reviewer: R.Huber Cited in 2 ReviewsCited in 88 Documents MathOverflow Questions: Why is the definition of the adic spectrum \(\operatorname{Spa}\,(A,A^+)\) the ”right” definition? MSC: 14G20 Local ground fields in algebraic geometry 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14L05 Formal groups, \(p\)-divisible groups 13J20 Global topological rings Keywords:affinoid rings; adic rings; \(f\)-adic ring; continuous valuations; formal schemes; rigid analytic variety × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] [B1] Bourbaki, N.: Topological vectorspaces. Berlin Heidelberg New York: Springer 1987 [2] [B2] Bourbaki, N.: commutative Algebra, Paris: Hermann 1972 [3] [BGR] Bosch, S., Güntzer, U., Remmert, R.: Non-archimedean analysis, Berlin Heidelberg New York: Springer 1984 · Zbl 0539.14017 [4] [EGA] Grothendieck, A., Dieudonne, J.: Eléments de Géométrie Algébrique III. Publ. Math., Inst. Hautes Études Sci.20 (1964) [5] [EGA] Grothendieck, A., Dieudonne, J.: Elément de Géométrie Algébrique I Berlin Heidelberg New York: Springer 1971 [6] [FP] Fresnel, J., Put, M. van der: Géométrie analytique rigide et applications. Boston Basel: Birkhäuser 1981 [7] [G] Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J.9, 119–221 (1957) · Zbl 0118.26104 [8] [H1] Huber, R.: Continuous valuations. Mathematische Z.212, 455–477 (1993) · Zbl 0788.13010 · doi:10.1007/BF02571668 [9] [H2] Huber, R.: Coherent sheaves on adic spaces. (in preparation) [10] [H3] Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. (pre-print) [11] [M] Mumford, D.: An analytic construction of degenerating abelian varieties over complete rings. Compos. Math.24, 239–272 (1972) · Zbl 0241.14020 [12] [R] Raynaud, M.: Géométrie analytique rigide d’apres Tate, Kiehl,... Mém. Soc. Math. Fr.39–40, 319–327 (1974) [13] [SGA] Artin, M., Grothendieck, A., Verdier, J.L.: Théorie des Topos et Cohomologie Étale des Schémas. Berlin Heidelberg New York: Springer 1972 [14] [T] Tate, J.: Rigid analytic spaces. Invent. Math.12, 257–289 (1971) · Zbl 0212.25601 · doi:10.1007/BF01403307 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.