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Notes on condensed mathematics

Kiran S. Kedlaya

Section 13 Localization for solid analytic rings

In this section, we indicate how the sheaf theory of Huber rings translates into the setting of solid analytic rings.
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Reference.

This section is (very loosely) based on [8], Lectures 9 and 10. See also [19], Lecture 1 (for the version of the story using Huber rings) and [1] (for the solid analogue).
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Subsection 13.1 On the formation of rational localizations

We break down the structure of rational localizations in somewhat more detail.
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Remark 13.1.1.

In the following discussion, at certain points we will need to assume that we are working with a solid condensed ring \(A\) which admits a topologically nilpotent unit \(q\text{.}\) When this is not the case, we can formally push ourselves back into that situation by replacing \(A\) with \(A \otimes_{\underline{\ZZ} \solid} \underline{\ZZ((q))}\text{,}\) noting that the map \(A \to A \otimes_{\underline{\ZZ} \solid} \underline{\ZZ((q))}\) is split in \(\Mod_A \times_{\CAb} \CAb_\solid\) by the constant coefficient map.
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We can make a similar construction starting with a Huber ring \(A\text{:}\) choose a ring of definition \(A_0\) and an ideal of definition \(I_0\text{,}\) complete \(A_0[q]\) with respect to the \((I_0,q)\)-adic topology, then tensor over \(A_0[q]\) with \(A[q^{\pm}]\text{.}\)
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Proof.

By rewriting
\begin{equation*} M \otimes^L_A A^{\triangleright} \langle T \rangle \cong M \otimes^L_A A^{\triangleright} \otimes^L_{\underline{\ZZ}\solid} \underline{\ZZ} \langle T \rangle \end{equation*}
we reduce to the case \(A = (\underline{\ZZ}, \Mod_{\underline{\ZZ} \solid})\text{.}\) Hereafter, we implicitly apply Theorem 7.4.1 to transfer results between \(\Ab_\solid\) and \(\CAb_\solid\text{.}\)
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We next pass through a sequence of reduction steps on \(M\text{.}\) By Proposition 2.4.8, we may assume that \(M\) is finitely presented. By Proposition 2.3.12, we have a splitting
\begin{equation*} M = \left( \prod_I \underline{\ZZ} \right) \oplus \iExt^1_{\underline{\ZZ}}(\underline{Q}, \underline{\ZZ}) \end{equation*}
where \(I\) is at most countable and \(Q\) is an at most countable abelian group, and we may treat the two factors separately. For the first factor, using the compatibility of solid tensor product with countable products, we may reduce to the case where \(I\) is a singleton, at which point the claim is apparent. For the second factor, using Remark 2.3.11 we may further reduce to the cases where \(Q\) is either a torsion group or a \(\QQ\)-vector space.
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Suppose that \(M = \iExt^1_{\underline{\ZZ}}(\underline{Q}, \underline{\ZZ})\) where \(Q\) is a torsion group. Using Proposition 2.3.8 and the compatibility of solid tensor product with countable limits, we may reduce to the case where \(Q\) is itself finite. By the usual structure theorem for finite abelian groups, we may further assume \(Q\) is even cyclic, say \(Q = \ZZ/m\ZZ\) and hence \(M \cong \underline{\ZZ/m\ZZ}\text{.}\) At this point, we must check that for any at most countable set \(I\text{,}\) an element of \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \langle T \rangle \right)\) is divisible by \(m\) if its image in \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \llbracket T \llbracket \right)\) is divisible by \(m\text{.}\) Now identify the elements of
\begin{equation*} \Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \llbracket T \rrbracket \right) = \prod_\NN \bigoplus_I \ZZ \end{equation*}
with \(\NN \times I\) matrices whose row vectors have finite support, and the elements of \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \langle T \rangle \right)\) with \(\NN \times I\) matrices whose row and column vectors have finite support. From this description, it is clear that dividing out a factor of \(m\) does not disturb any support conditions; alternatively, since we already start with a support condition on row vectors, we may reduce to the case where \(I\) is a singleton, which is apparent.
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Suppose that \(M = \iExt^1_{\underline{\ZZ}}(\underline{Q}, \underline{\ZZ})\) where \(Q\) is a \(\QQ\)-vector space. By Example 2.3.10, we then have \(M \cong \prod_J \underline{\widehat{\ZZ}/\ZZ}\) for some \(I\) with \(|J| \leq \kappa\text{;}\) we again reduce to the case where \(I\) is a singleton. At this point, we must check that for any at most countable set \(I\text{,}\) an element of \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\widehat{\ZZ}} \langle T \rangle \right)\) factors through \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \langle T \rangle \right)\) if its image in \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\widehat{\ZZ}} \llbracket T \rrbracket \right)\) factors through \(\Hom_{\underline{\ZZ}}\left( \prod_I \underline{\ZZ}, \underline{\ZZ} \llbracket T \rrbracket \right)\text{.}\) Using the matrix interpretation from the previous paragraph, we reduce to the case where \(I\) is a singleton, which is apparent.
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Proof.

Using Lemma 13.1.2, we reduce to the corresponding assertion for \(\prod_\NN M\text{,}\) on which multiplication by \(1-fT\) is an isomorphism.
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Proof.

The statement for \(M \otimes_A A^{\triangleright}[T]= \bigoplus_{n \in \NN} MT^n\) is obvious because \(T-f\) is a monic polynomial. For \(M \otimes_A A^{\triangleright}\langle T \rangle\text{,}\) we break the argument into various cases.
  • If \(M\) is a module over \(A^{\triangleright}[1/f]\text{,}\) then \(T-f\) and \(1-f^{-1} T\) have the same kernel on \(M \otimes_A A^{\triangleright}[T]_{\liquid T}\) and we may invoke Lemma 13.1.3.
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  • If \(M\) is \(f\)-torsion-free, then \(M\) embeds into \(M \otimes_A A^{\triangleright}[1/f]\) and the previous case applies.
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  • If \(M\) is killed by \(f^m\) for some positive integer \(m\text{,}\) then multiplication by \(T^m\) on \(M \otimes_A A^{\triangleright}[1/f]\) is visibly injective, but this coincides with multiplication by \(T^m-f^m = (T-f)(T^{m-1} + fT^{m-2} + \cdots + f^{m-1})\text{.}\) Consequently, multiplication by \(T-f\) is also injective.
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  • If \(M\) is \(f\)-torsion, then it is the colimit of its submodules killed by \(f^m\) over all \(m \in \NN\) and the previous case applies.
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  • Finally, in the general case we have an exact sequence
    \begin{equation*} 0 \to M[f^\infty] := \colim_n \ker(f^n, M) \to M \to M/M[f^\infty] \to 0 \end{equation*}
    in which \(M[f^\infty]\) is \(f\)-torsion and \(M/M[f^\infty]\) is \(f\)-torsion-free. Hence we may reduce to previously settled cases.
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Proof.

The tensor product in \(A\) is right exact by Hom-tensor adjunction, and exactness at the left is preserved by Lemma 13.1.4.
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We note some limitations of the previous discussion.
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Example 13.1.6.

By contrast with Corollary 13.1.5, the morphism \(\times (T-f)\) on \(A^{\triangleright}\langle T^{\pm} \rangle\) in \(\Mod_A\) is not necessarily injective, let alone universally injective. The following example is essentially that of Rost from [17], end of section 1; by replacing the base ring \(\ZZ\) with a nonarchimedean field one can make an example where \(A\) is a Tate Huber ring, which then reproduces [5], Proposition 12.
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Fix \(c \in (0,1)\text{,}\) let \(A^{\triangleright}\) be the completion of \(\ZZ[f^{\pm},Z]/(Z^2)\) for the submultiplicative norm
\begin{equation*} \left| \sum_{n \in \ZZ} a_n f^n + \sum_{n \in \ZZ} b_n f^n Z \right| = \max_{n \in \ZZ} \{|a_n| c^{-|n|}, |b_n| c^{|n|}\}\text{,} \end{equation*}
and take \(A^+\) to be the elements of norm at most \(1\text{.}\) Then \(A = (A^{\triangleright}, A^+)\) is a complete Huber ring and
\begin{equation*} x := \sum_{n \in \ZZ} f^{-n} Z T^n \end{equation*}
is a well-defined nonzero element of \(A^{\triangleright} \langle T^{\pm} \rangle\) such that \((T-f)x = 0\text{.}\)
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Remark 13.1.7.

Corollary 13.1.5 does not imply that \(A^{\triangleright} \langle f \rangle\) is a flat object of \(\Mod_A\) (meaning that tensoring with it is exact); indeed, the failure of flatness can be seen from Example 9.1.6. Rather, we only conclude that the connecting homomorphism out of the first Tor group is always zero. In particular, we deduce that \(A^{\triangleright} \langle T \rangle\) is a universally injective (by Lemma 13.1.2) but not flat object of \(\Mod_A\text{;}\) in particular, \(\ZZ \langle T \rangle\) is a universally injective but not flat object of \(\Mod_{\ZZ\solid}\text{.}\)
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Note also that this failure of flatness holds even when when \(A\) is a strongly noetherian Huber ring, in which case the map from \(A\) to any rational localization is flat in the algebraic sense ([17], proof of Theorem 2.5, case II, point (II.1.iv)). There is no contradiction here because the two notions of flatness concern different tensor product functors.
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One can interpret the phenomenon being observed here as the fact that a rational localization must be distinguished from a derived rational localization, which by fiat is resolved by a Koszul complex. In this way, we can also interpret Example 13.1.6 as the failure of a rational localization (in this case \(A \to A \langle f^{\pm} \rangle\)) to coincide with the corresponding derived rational localization.
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We will have more to say about this point in Section 15.
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Subsection 13.2 The adic topology and the structure presheaf

We next introduce a presheaf of rings associated to a solid analytic rings. This presheaf fails to be a sheaf in general, partially echoing the corresponding failure for Huber rings; see Subsection 13.3.
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Proof.

It suffices to treat the case \(R = \underline{\ZZ}\text{,}\) as we obtain the general case by taking a solid tensor product with \(R\) over \(\ZZ\text{.}\) We may treat that case in either \(\Ab_\solid\) or \(\CAb_\solid\) by virtue of Theorem 7.4.1; we do both for the sake of illustration.
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In \(\Ab_\solid\text{,}\) evaluating \(\underline{\ZZ}\langle T,T^{-1} \rangle\) at \(\prod_I \ZZ_\solid\) yields the intersection of \(\prod_I \ZZ[T,T^{-1}]\) and \(\prod_\ZZ \bigoplus_I \ZZ\) within \(\prod_\ZZ \prod_I \ZZ\text{.}\) This clearly splits into the two indicated summands.
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In \(\CAb_\solid\text{,}\) evaluating at \(S = \varprojlim_i S_i \in \Prof\) has the following effect: for \(R \langle T \rangle\) we get power series \(\sum_n a_n T^n\) in which the coefficients \(a_n\) are finite sums of \(\ZZ\)-valued Dirac measures with the property that for any fixed closed-open subset \(U\) of \(S\text{,}\) the integral of \(U\) against \(a_n\) is eventually zero. From the corresponding descriptions for \(R\langle T^{-1} \rangle\) and \(R\langle T^{\pm} \rangle\text{,}\) we infer the desired exact sequence.
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Proof.

We first observe that the sequences
\begin{gather*} 0 \to R \langle T \rangle \stackrel{\times f-gT}{\to} R\langle T \rangle \to R\left\langle \tfrac{f}{g} \right\rangle \to 0\\ 0 \to R \langle T^{-1} \rangle \stackrel{\times g-fT^{-1}}{\to} R \langle T^{-1} \rangle \to R\left\langle \tfrac{g}{f} \right\rangle \to 0 \end{gather*}
are exact. If \(g = 1-f\) then \(R\left\langle \tfrac{f}{g} \right\rangle = R\left\langle \tfrac{1}{1-f} \right\rangle\) and \(R\left\langle \tfrac{g}{f} \right\rangle = R\left\langle \tfrac{1}{f} \right\rangle\text{,}\) so both claims follow from Lemma 13.1.3. If \(g = 1\text{,}\) then we apply both Lemma 13.1.3 and Lemma 13.1.4.
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We then apply Lemma 13.2.1 to obtain an exact sequence
\begin{equation*} 0 \to 0 \to R\langle T \rangle \oplus T^{-1} R\langle T^{-1} \rangle \to R\langle T^{\pm} \rangle \to 0 \end{equation*}
and a split exact sequence
\begin{equation*} 0 \to R \to R\langle T \rangle \oplus R \langle T^{-1} \rangle \to R \langle T^{\pm} \rangle \to 0\text{.} \end{equation*}
Mapping the first sequence to the second via multiplication by \(f-gT = T(g-fT^{-1})\) and applying the snake lemma yields the desired seqeunce.
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Definition 13.2.3.

To make notation sensible, write \(\AnSpec A\) for the object of \(\AnRing^{\op}\) corresponding to \(A \in \AnRing\text{.}\) Define the adic topology on \(\AnRing^{\op}\) as the topology corresponding to coverings of \(\Spa A(*)\) where \(A(*)\) denotes the Huber pair \((A^{\triangleright}(*), A^+)\) for the discrete topology on \(A^{\triangleright}\text{.}\) Define the structure presheaf \(\calO\) on \(\AnRing^{\op}\) by \(\AnSpec A \mapsto A^{\triangleright}\text{.}\)
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Proof.

Definition 13.2.5.

We say that \(A \in \AnRing_\solid\) is sheafy if the structure presheaf is a sheaf for the adic topology. By Lemma 13.2.2 and Proposition 12.3.8, this is equivalent to saying that for every rational localization \(A \to B\) and every \(f \in B\text{,}\) \(T-f\) and \((1-f)T-f\) are not zero divisors on \(B^{\triangleright} \langle T^{\pm} \rangle\text{.}\) However, it is not enough to impose this condition only for \(B = A\text{.}\)
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Proof.

By Lemma 13.2.2, we have the vanishing of higher Cech cohomology for standard binary coverings defined by parameters of the form \(f,1\) or \(f,1-f\text{.}\) Using Proposition 12.3.8, we may parlay this into the vanishing of higher Cech cohomology for general coverings. The latter formally implies the vanishing of higher sheaf cohomology by a general spectral sequence argument (compare [28], Tag 01EW and [19], Lemma 1.6.3).
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Proof.

Remark 13.2.8.

Let \(A\) be a solid analytic ring for which the natural map \(A \to H^0(\AnSpec(A), \calO)\) fails to be injective. Then \(A' := H^0(\AnSpec(A), \calO)\) is again a solid analytic ring, but we do not know whether it is sheafy because sheafiness also includes a corresponding restriction on every rational localization of \(A\text{.}\)
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On the other hand, if \(A\) admits a covering by rational localizations \(A \to B_i\) such that each \(B_i\) is sheafy, then it can be shown using Theorem 13.2.6 that \(H^0(\AnSpec(A), \calO)\) is sheafy.
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Subsection 13.3 Sheafiness for Huber rings

We now introduce the structure presheaf of a Huber pair and study the extent to which it does or does not satisfy the sheaf property. This will reveal that there are two different failure modes for sheafiness of Huber rings which can be separated by working with solid analytic rings.
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Definition 13.3.1.

For \(A\) a Huber pair, the structure presheaf \(\calO\) on \(\Spa A\) assigns to every rational subspace \(U\left( \tfrac{f_1,\dots,f_n}{g} \right)\) the quotient of \(A\left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \right\rangle^{\triangleright}\) indicated in Remark 12.2.9. (It is then extended to arbitrary open subspaces by taking stalks.) We say \(A\) is locally Tate if \(A^{\circ \circ}\) generates the unit ideal in \(A^{\triangleright}\text{;}\) this is equivalent to saying that \(\Spa A\) admits a covering by rational subspaces each of whose associated Huber ring is Tate. (This condition is called analytic in [19], but that terminology clashes with the notion of an analytic ring being used throughout these notes.)
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We say \(A\) is sheafy if \(\calO\) is a sheaf. This is a property of the underlying Huber ring \(A^{\triangleright}\) (which must in particular be complete).
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Proof.

For the “if” assertion, we may deduce the claim from Definition 12.2.2 in order to compare the concepts of Tate algebras (and by extension rational localizations) in the Huber setting and the solid setting.
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For the “only if” assertion, by Proposition 12.3.8 we are reduced to reduce to checking that if \(\underline{A} \to B_1, \underline{A} \to B_2\) are the rational localizations in a standard binary covering generated by \(f,1\) or \(f,1-f\text{,}\) then firstly \(B_1, B_2\) represent complete Huber rings, and secondly
\begin{equation*} 0 \to A^{\triangleright} \to B_1^{\triangleright} \oplus B_2^{\triangleright} \to B_{12}^{\triangleright} \to 0 \end{equation*}
is an exact sequence of solid abelian groups. For the first assertion, we must check that if \(f \in A^\circ\text{,}\) then the multiplication maps by \(1-fT\) and \(f-T\) in \(A^{\triangleright} \langle T \rangle\) are strict inclusions; this can be seen by working in terms of graded rings, as in [19], Lemma 1.8.2. (The ultimate point is that in any univariate polynomial ring, an element cannot be a zero divisor if either its constant coefficient or its leading coefficient is a unit.) For the second assertion, by Lemma 13.2.2 we need only check exactness at the left; by the first assertion this equates to the corresponding exactness using Huber’s rational localizations, which follows from \(A\) being sheafy.
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Proof.

When \(A\) is Tate, this follows by combining Theorem 13.2.6 with Proposition 13.3.2 and the same spectral sequence argument as in the proof of Theorem 13.2.6.
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In the general case, we need a slightly different argument. We may apply Remark 13.1.1 to replace \(A^{\triangleright}\) with \(A^{\triangleright}((q))\text{,}\) but it does not formally follow that the latter is sheafy. Nonetheless, we may argue as in Proposition 13.3.2 that every rational localization of \(\underline{A^{\triangleright}((q))}\) which is a base extension from \(A\) does represent a complete Huber ring; we may then continue as in the previous paragraph.
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Remark 13.3.5.

The sheafy condition for Huber pairs is somewhat subtle; it is known to hold in certain classes (e.g., for strongly noetherian Huber rings in the sense of Example 11.1.4, and for perfectoid rings) and to fail in certain carefully constructed examples, but in most cases it is difficult to decide whether a given Huber pair is sheafy. See [13] for a thorough discussion.
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Subsection 13.4 Solid adic spaces

Although this discussion is aimed towards a study of analytic stacks, it may be useful for conceptual purposes to note that we can already define something analogous to the category of schemes.
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Definition 13.4.1.

Let \(\LRS\) denote the category of locally ringed spaces. An object of \(\LRS\) is a topological space \(X\) equipped with a sheaf of rings \(\calO_X\) with the property that for each \(x \in X\text{,}\) the stalk \(\calO_{X,x}\) is a local ring (a ring with a unique maximal ideal). A morphism \((Y, \calO_Y) \to (X, \calO_X)\) of \(\LRS\) consists of a continuous map \(f\colon Y \to X\) together with a ring homomorphism \(f^\sharp\colon \calO_X \to f_*\calO_Y\) which, for each \(y \in Y\) mapping to \(x \in X\text{,}\) induces a local homomorphism \(\calO_{X,x} \to \calO_{Y,y}\) of local rings (i.e., the maximal ideal contracts to the maximal ideal).
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Remark 13.4.2.

In \(\LRS\text{,}\) the affine scheme \(\Spec R\) corresponding to \(R \in \Ring\) has the following universal property: for every \(X \in \LRS\text{,}\) homomorphisms \(R \to \Gamma(X, \calO_X)\) correspond to morphisms \(X \to \Spec R\) via taking global sections. In other words, \(\Spec\) is right adjoint to the global sections functor \(\LRS \to \Ring^{\op}\text{.}\)
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Of course we can also construct other interesting objects in \(\LRS\text{:}\) topological manifolds, smooth (\(C^{\infty}\)) manifolds, real analytic manifolds, complex analytic manifolds, complex analytic varieties, adic spaces, etc. While the adjunction property of \(\Spec\) tells us exactly how these other objects map to affine schemes, it does not explain how they map to more general schemes. In fact this is quite an interesting matter!
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Definition 13.4.3.

Let \(\LCRS\) be the category of locally condensed ringed spaces as follows. An object of \(\LCRS\) is a topological space \(X\) equipped with a sheaf of condensed rings \(\calO_X\) with the property that for each \(x \in X\text{,}\) the stalk \(\calO_{X,x}(*)\) is a local ring (without topology). A morphism \((Y, \calO_Y) \to (X, \calO_X)\) of \(\LRS\) consists of a continuous map \(f\colon Y \to X\) together with a condensed ring homomorphism \(f^\sharp\colon \calO_X \to f_*\calO_Y\) which, for each \(y \in Y\) mapping to \(x \in X\text{,}\) induces a local homomorphism \(\calO_{X,x}(*) \to \calO_{Y,y}(*)\) of local rings.
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Let \(\LCRS_\solid\) be the subcategory of locally solid ringed spaces, i.e., the locally condensed ringed spaces \(X\) for which the sheaf takes values in solid condensed rings. By Theorem 13.2.6, for any sheafy solid analytic ring \(A\text{,}\) we may define an object \(\AnSpec A \in \LCRS_\solid\) with underlying topological space \(\Spa(A(*))\) (for the discrete topology on \(A^{\triangleright}(*)\)); we call this the (solid) adic spectrum of \(A\text{.}\)
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A solid adic space is a locally condensed ringed space admitting a open covering by subspaces each of which is the solid adic spectrum of some sheafy solid analytic ring.
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Example 13.4.4.

We obtain a solid adic space \(\PP^1_{\solid}\) by glueing \(\AnSpec \ZZ[T]_\solid\) and \(\AnSpec \ZZ[T^{-1}]_\solid\) along \(\AnSpec \ZZ[T,T^{-1}]_\solid\text{.}\)
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Note that \(\PP^1_{\solid}\) admits a natural map to \(\PP^1_\ZZ\text{,}\) induced by the maps \(\AnSpec \ZZ[T]_\solid \to \Spec \ZZ[T]\text{,}\) \(\AnSpec \ZZ[T^{-1}]_\solid \to \Spec \ZZ[T^{-1}]\text{.}\) We will study this map later in the context of analytification (Section 16).
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Example 13.4.5.

Let \(A\) be a solid analytic ring such that both \(A\) and \(A \langle T \rangle\) are sheafy. By base extension from the previous example, we get the analytic projective line \(\PP^1_A\) over \(A\text{.}\) Let \(\GG_{m,A}\) be the maximal open subspace of \(\PP^1_A\) on which \(T\) and \(T^{-1}\) are both defined (and inverse to each other).
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Now choose a topologically nilpotent unit \(q \in A^{\triangleright}(*)\text{.}\) Then the action of \(q^\ZZ\) on \(\GG_{m,A}\) is free, so we can quotient to form a new solid adic space \(\GG_{m,A}/q^\ZZ\text{,}\) the Tate elliptic curve over \(A\text{.}\) It can be shown that this construction can be “algebraized”; we will return to this point later in a more general setting.
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