nLab adic space




Adic spaces are the basic objects in Huber’s approach to non-archimedean analytic geometry. They are built by gluing valuation spectra? of a certain class of topological rings. Unlike Berkovich analytic spectra the points of adic spaces correspond to valuations of arbitrary rank, not only rank one. If a Berkovich space is corresponding to a separated rigid analytic space then it can be obtained as the largest Hausdorff quotient of the corresponding adic space.

The framework of adic spaces are used to build perfectoid spaces out of perfectoid rings.



Let (A,A +)(A,A^{+}) be a Huber pair, i.e. AA is a Huber ring and A +AA^{+}\subseteq A is a ring of integral elements. The adic spectrum Spa(A,A +)\mathrm{Spa}(A,A^{+}) is the set of equivalence classes of continuous valuations ||\vert\cdot\vert on AA such that |A +|1\vert A^{+}\vert\leq 1.

If xx is a valuation, and gAg\in A, we also suggestively write g|g(x)|g\mapsto\vert g(x)\vert for the valuation xx applied to gg. The topology on Spa(A,A +)\mathrm{Spa}(A,A^{+}) is the one generated by open sets of the form

{x:|f(x)||g(x)|0}\lbrace x:\vert f(x)\vert\leq\vert g(x)\vert\neq 0\rbrace

where f,gAf,g\in A.


  • The final object in the category of adic spaces is Spa(,)\mathrm{Spa}(\mathbb{Z},\mathbb{Z}).

  • The adic closed disc over p\mathbb{Q}_{p} is given by Spa(A,A +)\mathrm{Spa}(A,A^{+}) where A= pTA=\mathbb{Q}_{p}\langle T\rangle and A += pTA^{+}=\mathbb{Z}_{p}\langle T\rangle.

  • The adic open disc over p\mathbb{Q}_{p} is the generic fiber of Spa(A,A)Spa( p, p)\mathrm{Spa}(A,A)\to\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p}), where A= p[[T]]A=\mathbb{Z}_{p}[[T]].


  • R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, 1996. x+450 pp. (MR2001c:14046)

  • Sophie Morel, Adic spaces (pdf)

  • Torsten Wedhorn, Adic spaces (arXiv:1910.05934)

  • Brian Conrad, A brief introduction to adic spaces, PDF.

Last revised on August 7, 2023 at 19:23:39. See the history of this page for a list of all contributions to it.