Algebraic geometry is like ‘functional programming’ in geometry, spaces can be described and studied via their functions. For example, an affine variety is described by its coordinate ring, and a scheme is described by its structure sheaf of rings.
In this view we see ‘higher points’ that we don’t usually see in ordinary geometry: an integral scheme \(X\) has a generic point \(\eta\). It is enormously useful to take an object of \(X\) and study it at the generic point \(\eta\) fibre: If a property holds at \(\eta\), it often holds on a dense open subset of \(X\).
The language of formal schemes is useful for studying the ‘infinitesimal neighbourhood’ of a closed subvariety \(Y \subseteq X\). For example, if \(X = \operatorname{Spec} A\) is affine and \(Y = V(I)\) is defined by an ideal \(I \subseteq A\), then the formal completion of \(X\) along \(Y\) is given by the formal spectrum of the \(I\)-adic completion \(\hat{A}\) of \(A\). This allows us to study properties of \(X\) in a neighbourhood of \(Y\).
In terms of functions, \(I\) are the defining equations of \(Y\), then \(A/I\) is the algebra of functions that are forced to satisfy \(I\). If we instead take \(I^n\) we will obtain more ‘functions’ on the space that describe local coordinates ‘near’ \(Y\). The \(I\)-adic completion \(\hat{A}\) is the limit of these algebras of functions, and can be thought of as the algebra of functions defined in an infinitesimal neighbourhood of \(Y\).
In this formal schemes, the model spaces are formal spectra \(\mathrm{Spf}A\) of admissible topological rings (e.g. \(\mathbb{Z}[[T]]\), \(\mathbb{Z}_p\), \(I\)-adic completions of noetherian rings). The points of \(\mathrm{Spf}A\) correspond to open prime ideals of \(A\), which are in bijection with the prime ideals of \(A\) that do not contain the ideal of definition \(I\). Thus, \(\mathrm{Spf}A\) can be identified with the subspace of \(\operatorname{Spec} A\) consisting of those primes. This space can be given topological structure and a sheaf \(\mathcal{O}_X\) of topological rings.
In the case of complex analytic geometry, the model space are the vanishing locus of holomorphic functions on open subsets of \(\mathbb{C}^n\). If \(X\) is a finite type \(\mathbb{C}\)-scheme, one can analytify \(X\) to obtain a complex analytic space \(X^\mathrm{an}\), which is universal for maps from complex analytic spaces to \(X\) (in terms of functor of points, \(h'_X=h_{X^\mathrm{an}}\)). Any if \(X\) is a complex analytic variety embeddable into projective space, then it is an analytification of a complex projective variety with equivalent categories of coherent sheaves respecting cohomology (i.e. \(H^i(X,\mathcal{F}) \cong H^i(X^\mathrm{an},\mathcal{F}^\mathrm{an})\) for any coherent sheaf \(\mathcal{F}\) on \(X\)).
Moving from archimedean fields to non-archimedean fields, there is expected to have a good theory of analytic geometry over non-archimedean fields. Which rings \((X,\mathcal{O}_X)\) are the model spaces for such a theory?
Naive answer
\(X\) is an open subset of \(K^n\), \(\mathcal{O}_X\) the sheaf of continuous / locally analytic functions.
This is unlikely the natural choice as \(X\) and \(K\) are totally disconnected and there are too many functions, functions glue overly easily. Remember we want our theory to emulate complex analytic geometry, so we expect \(H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1})=K\), but this is not the case for naive choice (we get many more functions).
Tate’s rigid analytic spaces
The model spaces are \(\mathrm{Spm}A\) for an affinoid \(K\)-algebra \(A\). The topology on \(\mathrm{Spm}A\) is not ordinary topology induced form \(\overline{K}^n\), but a Grothendieck topology given by admissible opens and admissible covers.
A rigid analytic space over \(K\) is a pair \((X,\mathcal{O}_X)\) carrying a Grothendieck topology and a sheaf \(\mathcal{O}_X\) of \(K\)-algebras, such that \(X\) has an admissible cover by affinoid spaces \(\mathrm{Spm}A_i\) and \(\mathcal{O}_X|_{\mathrm{Spm}A_i} \cong \mathcal{O}_{\mathrm{Spm}A_i}\) (locally isomorphic to affinoid spaces).
Rigid analytic spaces are good and natural, but they have some drawbacks that suggest they lack some geometric information:
You can create bijection on points \(Y=U\coprod S \to X\) that is not an isomorphism. Let \(X=\mathrm{Spm}K\langle T\rangle\), \(S=K\langle T,T^{-1}\rangle\) the unit circle, \(U=X\setminus S\). Then \(Y=U\coprod S \to X\) is a bijection on points but not an isomorphism. This is because \(U,S\) is not an admissible cover of \(X\), despite \(U\) and \(S\) are admissible opens.
This suggests that there are points in \(X\) that we \(Y\) is missing.
Another shortcoming is that the category of rigid analytic spaces does not ‘generalize’ the category of schemes.
For these reasons, Huber developed the theory of adic spaces. The model spaces are \(\mathrm{Spa}(A,A^+)\) for a Huber pair \((A,A^+)\), where \(A\) is a Huber ring (a topological ring with an open adic subring) and \(A^+ \subseteq A\) is an open integrally closed subring contained in the subring of power-bounded elements \(A^\circ\). The points of \(\mathrm{Spa}(A,A^+)\) are equivalence classes of continuous valuations on \(A\) that are bounded by 1 on \(A^+\).