Skip to main content

Notes on condensed mathematics

Kiran S. Kedlaya

Section 12 Topology of Huber pairs

In this section, we recall the topology on the category of Huber rings, again adapting to the setting of solid analytic rings.

Reference.

This section is based on [8], Lectures 7, 9, and 10.

Subsection 12.1 Whither the Zariski topology?

As a warmup, let us think carefully about how one might arrive at the Zariski topology on an affine scheme, or better the correspondng Grothendieck topology on the category of affine schemes, given that one wants to end up reconstructing the usual theory of schemes.

Remark 12.1.1.

By Yoneda’s lemma, a scheme \(X\) can be reconstructed from its functor of points, i.e., the functor \(S \to \Hom_{\Sch}(S,X)\) on \(\Sch\text{.}\) Since every scheme is covered by affine schemes, it is enough to use the restriction of this functor to the category of affine schemes, which is equivalent to the opposite category of rings. In fact the term scheme is meant to suggest this: a scheme is a “blueprint” for turning rings into sets carrying some meaningful algebro-geometric structure.

Example 12.1.2.

Let us spell out the functor of points for mapping a test scheme \(X\) into \(X \to \PP^1_\ZZ\text{.}\) Since we have the canonical line bundle \(\calO(1)\) on \(\PP^1_\ZZ\text{,}\) \(f^* \calO(1)\) is a line bundle on \(X\text{.}\) If we write \(\PP^1_\ZZ = \Proj \ZZ[s_0, s_1]\text{,}\) then the two sections \(s_0, s_1\) of \(\calO(1)\) pull back to sections of \(f^* \calO(1)\) which generate this bundle. We can then reconstruct \(f\) uniquely from the data of the line bundle \(f^* \calO(1)\) and the sections \(f^*(s_0), f^*(s_1)\text{.}\)
Put another way, the data of the map \(f\) is equivalent to the data of the short exact sequence of sheaves of \(\calO_X\)-modules
\begin{equation*} 0 \to \calL_1 \to \calO_X \oplus \calO_X \to \calL_0 \to 0 \end{equation*}
in which \(\calL_0\) is invertible: the bundle \(\calL_0\) corresponds to \(f^* \calO(1)\) and the two maps \(\calO_X \to \calL_0\) correspond to the two sections \(f^*(s_0), f^*(s_1)\text{.}\)
If \(X = \Spec R\) is an affine scheme, we can again replace the data with a short exact sequence of \(R\)-modules:
\begin{equation*} 0 \to L_1 \to R \oplus R \to L_0 \to 0 \end{equation*}
in which \(L_0\) is invertible (projective of rank 1). In particular, this functor can be described in terms of the category of \(R\)-modules, which is equivalent to the category of quasicoherent sheaves on \(X\text{.}\)
Now suppose we want to think of \(\PP^1_\ZZ\) as itself being built out of the two affine schemes \(\Spec \ZZ[s_1/s_0]\) and \(\Spec \ZZ[s_0/s_1]\text{.}\) We then have to cover \(\Spec R\) with two localizations corresponding to the subsets on which \(s_0\) and \(s_1\) are respectively nonzero. If \(L_0\) is trivial, then we can identify \(s_0\) and \(s_1\) with elements of \(R\) that generate the unit ideal and take the maps
\begin{equation*} \Spec R[T]/(s_1 T - s_0) \to \Spec R, \qquad \Spec R[T]/(s_0 T - s_1) \to \Spec R\text{.} \end{equation*}
The analogue in the general case is to convert \(R \oplus R \to L_0\) into the closed immersion
\begin{equation*} \Spec R = \Proj_R \Sym_R(L_0) \to \Proj_R \Sym_R(R \oplus R) = \PP^1_R\text{,} \end{equation*}
take the fiber product with the inclusion of one of the basic open subsets \(\Spec R[T]\) corresponding to dehomogenizing \(\Sym_R(R \oplus R) \cong R[T_0, T_1]\) (i.e., mapping one variable to \(T\) and the other to \(1\)), and then project via \(\PP^1_R \to \Spec R\text{.}\)

Remark 12.1.3.

The key role played in Example 12.1.2 by the category of coherent sheaves on \(X\) is perhaps not surprising. After all, for a ring \(R\text{,}\) every affine scheme over \(\Spec R\) can be interpreted as an \(R\)-algebra object in \(\QCoh(\Spec R)\text{.}\) By the same token, building a satisfactory theory of quasicoherent sheaves on analytic rings will be crucial to the development of analytic stacks.
It may not be readily apparent that Example 12.1.2 tells us everything we need to know to make Zariski coverings of affine schemes, but that is in fact the case.

Definition 12.1.4.

For \(R\) a commutative ring, define a standard binary covering of \(\Spec R\) to be a covering of the form
\begin{equation*} \Spec R\left[\tfrac{f}{g}\right] \to \Spec R, \qquad \Spec R\left[\tfrac{g}{f}\right] \to \Spec R \end{equation*}
for some \(f,g \in R\) generating the unit ideal. In particular there must exist \(a,b \in R\) with \(af + bg = 1\text{,}\) so \(\frac{1}{g} = a\tfrac{f}{g}+b\) and so \(R\left[\tfrac{f}{g}\right] = R\left[ \tfrac{1}{g} \right]\text{;}\) consequently, \(\Spec R\left[\tfrac{f}{g}\right] \to \Spec R\) induces an isomorphism \(\Spec R\left[ \tfrac{f}{g} \right] \cong U_g\) where \(U_g \subseteq \Spec R\) is the set of prime ideals not containing \(g\text{.}\)

Remark 12.1.6.

In scheme theory, it is obvious that a distinguished open subspace of a distinguished open subspace is again a distinguished open subspace. That is, if I start with a ring \(R\text{,}\) an element \(f \in R\text{,}\) and a second element \(g \in R[\tfrac{1}{f}]\text{,}\) then \(\Spec R[\tfrac{1}{f}][\tfrac{1}{g}] \to \Spec R\) induces an isomorphism onto \(U_{h}\) for some \(h \in R\text{;}\) namely, write \(g = g_1/f\) and take \(h := fg_1\text{.}\)
The situation is a bit more complicated for Huber rings; see Remark 12.2.8.

Proof.

For \(R \in \Ring\text{,}\) we have a basis of the Zariski topology on \(\Spec R\) given by the open subsets \(U_{f} = \Spec R[\tfrac{1}{f}]\) for \(f \in R\text{.}\) Given an open covering by \(U_{f_i}\) for some \(f_i \in R\text{,}\) the ideal generated by the \(f_i\) must not be contained in any prime ideal, or even any maximal ideal, so it must be the unit ideal; we can thus find some finite subset \(f_1,\dots,f_n\) of the \(f_i\) such that the original covering is refined by the covering by \(U_{f_1},\dots,U_{f_n}\text{,}\) and some \(a_1,\dots,a_n \in R\) such that \(a_1 f_1 + \cdots + a_n f_n = 1\text{.}\)
We prove by induction on \(n\) that this covering is refined by some covering generated by standard binary coverings. When \(n=1\text{,}\) \(f_1\) must be a unit and so we are considering the one-element covering, which we may ignore. When \(n=2\text{,}\) we have \(U_{a_1 f_1} \subseteq U_{f_1}\text{,}\) \(U_{a_2 f_2} \subseteq U_{f_2}\) and so the covering is refined by the standard binary covering defined by \(a_1 f_1\text{.}\)
When \(n \gt 2\text{,}\) set
\begin{equation*} g := a_1 f_1 + \cdots + a_{\lfloor n/2 \rfloor} = 1 - a_{\lfloor n/2 \rfloor+1} f_{\lfloor n/2 \rfloor} - \cdots - a_n f_n \end{equation*}
so that \(V_1 := \Spec R[\tfrac{1}{g}]\text{,}\) \(V_2 := \Spec R[\tfrac{1}{1-g}]\) form a standard binary covering of \(\Spec R\text{.}\) In the ring \(R[\tfrac{1}{g}]\text{,}\) the elements \(f_i\) for \(1 \leq i \leq \lfloor n/2 \rfloor\) generate the unit ideal; in the ring \(R[\tfrac{1}{1-g}]\text{,}\) the elements \(f_i\) for \(\lfloor n/2 \rfloor + 1 \leq i \leq n\) generate the unit ideal. Consequently, the original covering is refined by the covering by
\begin{equation*} \Spec R[\tfrac{1}{g}][\tfrac{1}{f_i}] \quad (i=1,\dots,\lfloor n/2 \rfloor), \quad \Spec R[\tfrac{1}{1-g}][\tfrac{1}{f_i}] \quad (i=\lfloor n/2 \rfloor+1, \dots,n)\text{.} \end{equation*}
Since both \(\lfloor n/2 \rfloor\) and \(n - \lfloor n/2 \rfloor\) are strictly less than \(n\text{,}\) we may apply the induction hypothesis to conclude.

Subsection 12.2 Rational localizations of Huber pairs

We next recall the construction of localization maps in Huber’s theory, in the language of solid analytic rings. We first provide some context from the previous discussion. (We will introduce further context when we discuss analytification more thoroughly in Subsection 16.1).

Remark 12.2.1.

For \(R\) a ring, we can formally reinterpret maps from \(\Spec R\) to \(\PP^1_\ZZ\) as maps from \(\Spec R\) to \(\PP^1_R = \PP^1_\ZZ \times_\ZZ R\) in the category of schemes over \(\Spec R\text{.}\) For \(R\) an analytic ring, we will want to consider maps from its associated geometric object to the analytic projective line over \(R\text{,}\) which means we should be able to pull back coverings of the latter.
For Huber pairs (or more generally solid analytic rings), we will want there to be a covering of the analytic projective line by two subspaces defined by the conditions \(|s_0| \leq |s_1|\) and \(|s_1| \leq |s_0|\text{,}\) corresponding in complex geometry to the covering of the Riemann sphere by the closed unit disc and its reciprocal. As for schemes (Proposition 12.1.7), this desire ends up completely dictating the topology of adic spectra (Proposition 12.3.8).
In order to explain Remark 12.2.1, we first construct the analogue of localization maps \(R \to R_f\) for complete Huber pairs, and more generally for solid condensed rings.

Definition 12.2.2.

Following Example 9.1.4, for any positive integer \(n\text{,}\) define the solid Tate algebra \(\underline{\ZZ}\langle T_1,\dots,T_n \rangle\) over \(\underline{\ZZ}\) as the solid condensed ring corresponding to the functor in \(\Ab_\solid\) given by
\begin{equation*} \prod_I \ZZ_\solid \mapsto \left( \prod_I \ZZ[T_1,\dots,T_n] \right) \cap \left( \bigoplus_I \ZZ \right) \llbracket T_1,\dots,T_n \rrbracket \end{equation*}
where the intersection is taken inside \(\left( \prod_I \ZZ \right) \llbracket T_1,\dots,T_n \rrbracket\text{.}\)
For any solid condensed ring \(R\text{,}\) set \(R \langle T_1,\dots,T_n \rangle := R \otimes_\solid \underline{\ZZ} \langle T_1,\dots,T_n \rangle\text{.}\) When \(R\) is the solid ring associated to a Tate Huber ring, this coincides with the usual definition of the Tate algebra (the ring of power series whose coefficients form a null sequence) by some computations as in Example 9.1.4 and Example 9.1.5.

Proof.

It suffices to check the case \(n=1\) and then induct; that is, we must check that
\begin{equation*} R[T]_{\liquid T} \cong R \langle T \rangle\text{.} \end{equation*}
This further reduces to the universal case \(R = \underline{\ZZ((q))}\) for the \(q\)-adic topology on \(\ZZ((q))\text{.}\) As noted in Definition 12.2.2, we may identify \(R \langle T \rangle\) with the usual notion of the Tate algebra over the Huber ring \(\ZZ((q))\text{,}\) and then it is straightforward to compare the two.

Remark 12.2.4.

In Lemma 12.2.3, the hypothesis that \(A\) admits a topologically nilpotent unit is indeed necessary. For instance, for \(A = \underline{\ZZ}\text{,}\) \(A[T]_{\liquid T} = \underline{\ZZ[T]}\) for the discrete topology, which is not equal to \(\ZZ \langle T \rangle\) as per Example 9.1.4. The point is that in this case, the passage from \(A[T]\) to \(A[T]_{\liquid T}\) is not sensitive enough to distinguish between \(\underline{\ZZ}[T]\) and \(\underline{\ZZ} \langle T \rangle\text{.}\)

Definition 12.2.5.

For \(A^{\triangleright}\) a solid condensed ring and \(f_1,\dots,f_n,g \in A^{\triangleright}(*)\text{,}\) define
\begin{equation} A^{\triangleright}\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \rangle := \frac{A^{\triangleright}\langle T_1,\dots,T_n \rangle}{(f_1-gT_1,\dots,f_n-gT_n)} \left[ \frac{1}{g} \right]\text{.}\tag{12.1} \end{equation}
For \(A\) a solid analytic ring, we correspondingly form a homomorphism \(A \to A \langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \rangle\) by setting
\begin{equation*} \Mod_{A \langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \rangle} := \Mod_{A^{\triangleright}\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \rangle \liquid T_1,\dots,T_n} \times_{\Mod_{A^{\triangleright}}} \Mod_A\text{.} \end{equation*}
Any homomorphism as above in which \(f_1,\dots,f_n,g\) generate the unit ideal will be called a rational localization of the solid condensed ring \(A^{\triangleright}\) or of the solid analytic ring \(A\text{.}\)
For comparison with Huber’s theory, we will also need to consider the case where \(f_1,\dots,f_n,g\) generate an ideal whose radical contains \(A^{\circ \circ}\text{.}\) In this case we refer to the resulting homomorphism as a weak rational localization.

Remark 12.2.6.

In Definition 12.2.5, note that nothing changes if we insert an extra copy of \(g\text{,}\) as \(A^{\triangleright} \langle T \rangle/(T-1) \cong A^{\triangleright}\text{.}\) This is convenient for some arguments.
For example, if \(f_1,\dots,f_n,g\) and \(f'_1,\dots,f'_m,g'\) both generate the unit ideal, then we can define a rational localization with parameters \(T \cup \{gg'\}\) where
\begin{equation*} T = \{t_1 \cdot t_2 \colon t_1 \in \{f_1,\dots,f_n,g\}, t_2 \in \{f'_1,\dots,f'_m,g'\}\} \end{equation*}
also generates the unit ideal. The resulting object is the tensor product of the two original rational localizations.

Remark 12.2.7.

When considering a rational localization as in Definition 12.2.5, \(1/g\) can be written as an \(A^{\triangleright}\)-linear combination of \(1, f_1/g, \dots, f_n/g\text{;}\) consequently, it is not necessary to invert \(g\) separately in (12.1). That is,
\begin{equation} A^{\triangleright} \left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g}\right\rangle = \frac{A^{\triangleright}\langle T_1,\dots,T_n \rangle}{(f_1-gT_1,\dots,f_n-gT_n)}\text{.}\tag{12.2} \end{equation}
The distinction between rational localizations and weak rational localizations collapses when \(A^{\triangleright}\) contains a topologically nilpotent unit, or more generally when \(A^{\circ \circ}\) generates the unit ideal of \(A\text{.}\) An example where the two concepts differ would be with \(A^{\triangleright} = \ZZ[x,y]\) with the \((x,y)\)-adic topology and the parameters \(x,y\text{.}\)
One reason for using weak rational localizations is that they have better behavior with respect to composition; see Remark 12.2.8. However, rational localizations are enough to generate the right topology, and it is somewhat easier to analyze coverings made solely of rational localizations; so we will prefer them in what follows.

Remark 12.2.8.

In contrast with Remark 12.1.6, it is not clear that a composition \(A \to B \to C\) of rational localizations is a rational localization. One issue is that the parameters defining \(B \to C\) do not belong to \(A\text{,}\) but this is easily remedied by replacing them with suitable approximations in \(A[\tfrac{1}{g}]\) and then clearing denominators. A much more serious issue is that the resulting parameters do not generate the unit ideal.
For weak rational localizations of a Huber ring, the composition is again a weak rational localization ([17], Lemma 1.5(ii)); as a corollary, if \(A \to B\) and \(A \to C\) are weak rational localizations, then the composition \(A \to B \to B \otimes_A C\) is again a weak rational localization. However, these arguments depend on the fact that a Huber ring contains a finitely generated ideal of definition, a property which has no analogue for solid analytic rings.
In any case, we will formulate a more intrinsic version of the rational localization property that will be automatically compatible with compositions, that of an idempotent morphism (Definition 15.3.1).

Remark 12.2.9.

When constructing a rational localization using Definition 12.2.5 starting with a Tate Huber pair \(A\) (more precisely, with an element of the essential image of the functor Proposition 11.3.13), it can still happen that the resulting solid analytic ring \(A^{\triangleright} \left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g}\right\rangle\) is not a Huber pair! The issue is that to get the object in the category of Huber pairs with the corresponding universal property, one must take the completion of the ideal \((f_1 - gT_1,\dots,f_n - gT_n)\) as this ideal may itself not be closed.
This failure turns out to be intimately related to the failure of general Tate Huber rings to be sheafy. We will discuss this in some detail in Section 13; see especially Proposition 13.3.2.

Subsection 12.3 Adic spectra of Huber rings

Definition 12.3.1.

Let \(R\) be a ring. A valuation on \(R\) is a function \(v\colon R \to \Gamma \cup \{0\}\text{,}\) where \(\Gamma\) is a totally ordered abelian group written multiplicatively, such that:
  • \(v(0) = 0\text{,}\) \(v(1) = 1\text{.}\) (We do not require \(v^{-1}(0) = \{0\}\text{.}\))
  • \(v(x+y) \leq \max\{v(x), v(y)\}\) for all \(x,y \in A^{\triangleright}\text{.}\)
  • \(v(xy) = v(x) v(y)\) for all \(x,y \in A^{\triangleright}\text{.}\)
We declare two valuations to be equivalent if they give rise to the same sets \(\{(x,y) \in R \times R \colon v(x) \leq v(y)\}\text{.}\) Define the valuative spectrum \(\Spv(R)\) to be the set of equivalence classes of valuations of \(R\text{,}\) equipped with the topology generated by subsets of the form \(\{v \in \Spv(R)\colon v(x) \leq v(y)\neq 0\}\) for \(x,y \in R\text{.}\) This is a spectral space ([16], Proposition 2.2).
We incorporate [16], Lemma 2.5(ii) in the following definition.

Definition 12.3.2.

Let \(A^{\triangleright}\) be a Huber ring. Let \(\Spv(A^{\triangleright}, A^{\circ \circ})\) be the subspace of \(\Spv(A^{\triangleright})\) consisting of those valuations \(v\) with the following property: either there is no convex proper subgroup of \(\Gamma_v := v(v^{-1}(\Gamma))\) containing \(\{v(a)\colon a \in A^{\triangleright}, v(a) \geq 1\}\text{;}\) or \(v(x)\) is cofinal in \(\Gamma_v\) for each \(x \in A^{\circ \circ}\text{.}\) Unpacking terminology here, a subgroup \(H\) of \(\Gamma\) is convex if for all \(x \leq y \in H\text{,}\) we have \([x,y] \subseteq H\text{;}\) and \(v(x)\) is cofinal in \(\Gamma_v\) if
\begin{equation*} \Gamma_v = \bigcup_{n=1}^\infty \Gamma_v \cap [v(x)^n, \infty)\text{.} \end{equation*}
(Note that we are not requiring that \(v(x) \in H\text{.}\)) When \(A^{\triangleright}\) is Tate, cofinality can be tested using a single topologically nilpotent unit \(x\text{.}\) The space \(\Spv(A^{\triangleright}, A^{\circ \circ})\) is again spectral ([16], Proposition 2.6(i)).
A valuation \(v\) is continuous if for every \(\gamma \in \Gamma_v\text{,}\) there is a neighborhood \(U\) of \(0\) in \(A^{\triangleright}\) such that \(v(x) \lt \gamma\) for all \(x \in U\text{.}\) Equivalently, \(v \in \Spv(A, A^{\circ \circ})\) and \(v(a) \lt 1\) for all \(a \in A^{\circ \circ}\) ([16], Theorem 3.1).
Let \(A = (A^{\triangleright}, A^+)\) be a Huber pair. The adic spectrum of \(A\) is the subspace \(\Spa(A) \subseteq \Spv(A^{\triangleright}, A^{\circ \circ})\) consisting of those equivalence classes of continuous valuations \(v\) for which \(A^+ \subseteq \{a \in A^{\triangleright}\colon v(a) \leq 1\}\text{.}\) This is again a spectral space ([16], Theorem 3.5(i)); moreover, the inclusion \(\Spa(A) \subseteq \Spv(A^{\triangleright}, A^{\circ \circ})\) has a canonical retraction (see the proof of Lemma 12.3.4).

Definition 12.3.3.

Let \(A = (A^{\triangleright}, A^+)\) be a Huber pair. For \(f_1,\dots,f_n,g \in A^{\triangleright}\) which generate the unit ideal, let \(U(\tfrac{f_1,\dots,f_n}{g})\) be the set of \(v \in \Spa(A)\) such that \(v(g) \neq 0\) and \(v(f_i) \leq v(g)\) for \(i=1,\dots,n\text{.}\) Any subspace of this form is called a rational subspace of \(\Spa(A)\text{;}\) these subspaces are open and form a basis of the topology (Lemma 12.3.4).
The rational localization \(A \to A^{\triangleright}\left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g}\right\rangle\) induces a homeomorphism \(\Spa A^{\triangleright}\left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g}\right\rangle \cong U(\tfrac{f_1,\dots,f_n}{g})\) ([17], Lemma 1.5(ii)). Note that this identification makes crucial use of the rings of integral elements!

Proof.

Consider a point \(v \in \Spa A\) contained in an open subspace \(U\) of \(\Spa A\text{.}\) Let \(c\Gamma_v\) be the convex subgroup of the value group \(\Gamma_v\) generated by all \(v(a) \geq 1\) for \(a \in A^{\triangleright}\text{;}\) we get a new valuation \(w \in \Spa A\) by setting \(w(a) = v(a)\) if \(v(a) \in c\Gamma_v\) and \(w(a) = 0\) otherwise. Now choose a rational subspace \(U(\tfrac{f_1,\dots,f_n}{g}) \subseteq U\) containing \(w\text{.}\) Since \(w(g) \neq 0\text{,}\) there exists \(a \in A\) with \(w(ag) \geq 1\text{.}\) Then \(U(\tfrac{1,af_1,\dots,af_n}{ag})\) contains both \(w\) and its generization \(v\text{.}\)

Definition 12.3.5.

For \(A\) a Huber pair, define a standard binary covering of \(\Spa A\) to be a covering of the form
\begin{equation*} U(\tfrac{f}{g}), U(\tfrac{g}{f}) \end{equation*}
for some \(f,g \in A\) generating the unit ideal.

Example 12.3.6.

As in the proof of Proposition 12.1.7, we will need standard binary coverings with \(g = 1-f\text{.}\) Unlike in the proof of Proposition 12.1.7, we will also need standard binary coverings with \(g = 1\text{;}\) these are sometimes called standard Laurent coverings. Note that in both cases, the terms in such a covering are all of the form indicated in Lemma 12.3.4.

Proof.

By Lemma 12.3.4, we may start with a covering of \(\Spa A\) by rational subspaces of the form \(U_i = U(\tfrac{1,f_{i1},\dots,f_{in_i}}{g_i})\) for \(i=1,\dots,m\text{.}\) Note that \(g_1,\dots,g_m\) must generate the unit ideal.
Let \(S\) be the set of products \(t_1 \cdots t_m\) with \(t_i \in \{1,f_{i1},\dots,f_{in_i},g_i\}\) for all \(i\) and \(t_i = g_i\) for at least one \(i\text{.}\) Since \(g_1,\dots,g_m \in S\text{,}\) \(S\) generates the unit ideal, so the sets \(U(\tfrac{S}{s})\) for \(s \in S\) do form a covering.
Finally, note that for each \(s = t_1 \cdots t_m \in S\text{,}\) if we choose an index \(i\) for which \(t_i = g_i\text{,}\) then we also obtain elements of \(S\) by replacing \(g_i\) with each of \(1,f_{i1},\dots,f_{in_i}\) in the product. This implies that \(U(\tfrac{S}{s}) \subseteq U_i\text{,}\) so we have found a refinement of our initial covering.

Proof.

Let \(A\) be a Huber pair. To prove that every covering of \(\Spa A\) is refined by a covering generated by standard binary coverings, thanks to Lemma 12.3.7, we are free to start with a covering of \(\Spa A\) by the sets \(U_i = U(\tfrac{f_1,\dots,f_n}{f_i})\) for \(i=1,\dots,n\) for some elements \(f_1,\dots,f_n \in A^{\triangleright}\) generating the unit ideal. We can then proceed by induction on \(n\text{,}\) with induction step as in the proof of Proposition 12.1.7. As for the base cases, note that \(n=1\) is trivial. For \(n=2\text{,}\) choose \(a_1, a_2 \in A^{\triangleright}\) with \(a_1 f_1 + a_2 f_2 = 1\text{,}\) then pass to the terms of the standard binary covering defined by \(a_1 f_1\text{;}\) on each term, at least one of \(f_1\) or \(f_2\) is a unit and so we are considering a standard Laurent covering.

Remark 12.3.9.

For Tate Huber pairs, one can further refine Proposition 12.3.8 to say that only standard Laurent coverings are needed. Namely, given \(f,q \in A^{\triangleright}\) with \(q\) a topologically nilpotent unit, the subspace \(U(\tfrac{1-f}{f}) = U(\tfrac{1}{f})\) contains \(U(\tfrac{1}{fq^n})\) for \(n\) sufficiently large; so we can refine any standard binary covering by a composition of standard Laurent coverings.

Remark 12.3.10.

Proposition 12.3.8 suggests that to topologize the category of analytic rings, it will already give useful results to just take the coverings generated by standard binary coverings. We will pursue this point in Section 13.
PrevTopNext