Localization of Ringed Locales

Question

Let$\newcommand{\from}{\colon}\newcommand{\Spec}{\mathrm{Spec}}\newcommand{\sheaf}{\mathscr}\newcommand{\Ouv}{\mathrm{Ouv}}\newcommand{\restr}[1]{|_{#1}}\newcommand{\loc}{{\mathrm{loc}}}\newcommand{\Et}{\mathrm{Ét}}$ $X$ be a locale and $\sheaf{O}_X$ a sheaf of rings on $X$. The ringed locale $(X, \sheaf{O}_X)$ is called locally ringed if for any open $U \in \Ouv(X)$ and sections $f_1, \dots, f_r \in \sheaf{O}_X(U)$ such that $f_1 + \dots + f_r$ is a unit, there exists a covering $U = V_1 \cup \dots \cup V_r$ such that each $f_i\restr{V_i}$ is invertible. It is easy to check that this agrees with the terminology for ringed topological spaces. My question is whether the forgetful functor $$\{\text{Locally ringed locales}\} \to \{\text{Ringed locales}\}$$ admits a right adjoint $$\{\text{Ringed locales}\} \to \{\text{Locally ringed locales}\}, \qquad (X, \sheaf{O}_X) \mapsto (X_\loc, \sheaf{O}_{X_\loc}).$$

I would prefer to have a description of $(X_\loc, \sheaf{O}_{X_\loc})$ which is as down-to-earth as possible.

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If $X$ is the terminal locale (the discrete locale on a one-point set), then the sheaf $\sheaf{O}_X$ is the constant sheaf associated to a ring $R$, and then $X_{\loc}$ is the spectrum $\Spec(R)$ with the frame of opens given by $\Ouv(X_\loc) = \{\text{Radical ideals of }R\}$, $\sheaf{O}_{X_\loc}$ is the structure sheaf of the affine scheme $\Spec(R)$. Equivalently, we have $$\Ouv(X_\loc) = \{\text{Sheaves of radical ideals of } \sheaf{O}_X\}\qquad(*)$$ in this case.

The whole questions reminds me a little bit of the construction of the locale étalé associated to a pair $(X, \sheaf{F})$ where $X$ is a locale and $\sheaf{F}$ a sheaf of sets on $X$. In that case the frame of opens of $\Et(\sheaf{F})$ is given by the ordered set of subsheaves of $\sheaf{F}$. (See Chap. 2 of [F. Borceux, Handbook of categorical algebra. 3: Categories of sheaves. Cambridge: Cambridge Univ. Press (1994; Zbl 0911.18001)].) So one could try to use $(*)$ as a definition also in the general case.

However it is easy to see that if $X$ is already a locally ringed locale, then the projection $\pi_X \from (X_\loc, \sheaf{O}_{X_\loc}) \to (X, \sheaf{O}_X)$ should be an isomorphism. But if $X$ is a scheme, it is well known that $(*)$ does not produce an isomorphism, instead one should put $$\Ouv(X_\loc) = \{\text{Quasi-coherent sheaves of radical ideals of }\sheaf{O}_X\}.$$ But unfortunately, I am not sure what quasi-coherence should mean for a general ringed locale. I have heard that part of the goals of Scholze's condensed mathematics is to make sense of quasi-coherent sheaves on complex or non-archimedean analytic spaces, where clasically one only has a well-behaved theory of coherent sheaves.

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The analogous question whether the functor $$\{\text{Locally ringed spaces}\} \to \{\text{Ringed spaces}\}$$ has a right adjoint, has a positive answer: The right adjoint maps a ringed space $(X, \sheaf{O}_X)$ to the locally ringed space $(X_\loc, \sheaf{O}_{X_\loc})$ where $X_\loc = \{(x, \mathfrak{p}_x) \mid x \in X, \mathfrak{p}_x \text{ prime ideal of } \sheaf{O}_{X, x}\}$ together with a suitable topology and structure sheaf. See Cor. 6 of [W. D. Gillam, Adv. Pure Math. 1, No. 5, 250--263 (2011; Zbl 1247.14001)]. As Martin remarks in the comments, if one had a description of the frame of open subsets of Gillam's construction, one could mimic the construction in the localic setting.

Answer 1

I think the most down-to-earth approach is to make the classical $\operatorname{Spec}$ construction constructive, interpret it in the internal logic of the topos, then externalise the result. This is basically what you guessed – just take the frame of radical ideals of the ring – and is explained in § 12.4 of the PhD thesis of Ingo Blechschmidt. Note especially that it is not the same thing as the relative $\operatorname{\underline{Spec}}$, which has a different universal property.

But if you only care about existence, then there is a more conceptual approach. A ringed (Grothendieck) topos is basically the same thing as a topos equipped with a geometric morphism to the classifying topos for (commutative) rings, i.e. $\mathbf{B} (\textbf{CRing}) = \textbf{Psh} (\textbf{CRing}_\textrm{fp}^\textrm{op})$. We even get the correct notion of morphism for free: it is a geometric morphism between the toposes together with a 2-morphism from the composite geometric morphism to the other one. Diagrammatically: $$\require{AMScd} \begin{CD} \mathcal{Y} @>{\ulcorner O_\mathcal{Y} \urcorner}>> \mathbf{B} (\textbf{CRing}) \\ @VVV \Rightarrow @| \\ \mathcal{X} @>>{\ulcorner O_\mathcal{X} \urcorner}> \mathbf{B} (\textbf{CRing}) \end{CD}$$

A locally ringed topos is basically the same thing as a topos equipped with a geometric morphism to the Zariski topos, i.e. $\mathbf{B} (\textbf{LCRing}) = \textbf{Sh} (\textbf{CRing}_\textrm{fp}^\textrm{op}, \textrm{Zar})$. This gives the right objects, but unfortunately the natural notion of morphism suggested by this characterisation ends up being the same thing as a morphism of ringed toposes, contrary to the conventional definition. This is probably the root cause of the failure of obvious/naïve constructions: for example, taking the pullback of the classifying morphism $\ulcorner O_\mathcal{X} \urcorner : \mathcal{X} \to \mathbf{B} (\textbf{CRing})$ and the inclusion $\mathbf{B} (\textbf{LCRing}) \hookrightarrow \mathbf{B} (\textbf{CRing})$ gives the maximal subtopos of $\mathcal{X}$ such that the restriction of $O_\mathcal{X}$ is a local ring, and taking the comma object instead gives the classifying topos for $O_\mathcal{X}$-algebras, which is also not what we want.

Nonetheless, the yoga of classifying toposes suggests a way forward. The universal ring in $\mathbf{B} (\textbf{CRing})$ "is" simply the inclusion $\textbf{CRing}_\textrm{fp} \to \textbf{CRing}$. If the prime spectrum is truly a natural construction, then it should be possible to relativise it to rings in arbitrary toposes by constructing it once for all for the universal ring. There is an obvious candidate: take the classical prime spectrum functor $\textbf{CRing}_\textrm{fp} \to \textbf{Frm}$, then use the Grothendieck construction to obtain a site fibred in frames over $\textbf{CRing}_\textrm{fp}^\textrm{op}$. Somewhat more explicitly, the underlying category has objects pairs $(A, U)$ where $A$ is a finitely presented ring and $U$ is an open subspace of $\operatorname{Spec} A$, morphisms $(A, U) \to (B, V)$ are ring homomorphisms $\phi : B \to A$ such that the image of $U$ under $\operatorname{Spec} \phi : \operatorname{Spec} A \to \operatorname{Spec} B$ is a subspace of $V$, and the Grothendieck topology is the one generated by declaring $\{ (B, V_\alpha) \to (B, V) \}$ to be covering if the underlying ring homomorphisms are all $\textrm{id}_B$ and $\bigcup_\alpha V_\alpha = V$. Let $\mathbf{B} (\textbf{CRingP})$ be category of sheaves on this site. By construction, the forgetful functor induces a localic geometric morphism $\mathbf{B} (\textbf{CRingP}) \to \mathbf{B} (\textbf{CRing})$, and we also have a local ring in $\mathbf{B} (\textbf{CRingP})$, namely the sheaf $(A, U) \mapsto O_{\operatorname{Spec} A} (U)$, giving a geometric morphism $\mathbf{B} (\textbf{CRingP}) \to \mathbf{B} (\textbf{LCRing})$.

Define $\mathop{\textbf{Spec}} O_\mathcal{X}$ by the following pullback diagram (in the bicategory of toposes): $$\begin{CD} \mathop{\textbf{Spec}} O_\mathcal{X} @>>> \mathbf{B} (\textbf{CRingP}) \\ @VVV @VVV \\ \mathcal{X} @>>{\ulcorner O_\mathcal{X} \urcorner}> \mathbf{B} (\textbf{CRing}) \end{CD}$$ Localic geometric morphisms are closed under pullback, so $\mathop{\textbf{Spec}} O_\mathcal{X}$ is localic over $\mathcal{X}$; in particular, if $\mathcal{X}$ is localic, so is $\mathop{\textbf{Spec}} O_\mathcal{X}$. In any case, $\mathop{\textbf{Spec}} O_\mathcal{X}$ is a ringed topos in two (possibly different) ways: first, pulling back either $O_\mathcal{X}$ from $\mathcal{X}$ or $(A, U) \mapsto A$ from $\mathbf{B} (\textbf{CRingP})$ gives a (possibly non-local) ring $A_{\mathop{\textbf{Spec}} O_\mathcal{X}}$, but more importantly, pulling back the local ring from $\mathbf{B} (\textbf{CRingP})$ gives a local ring $O_{\mathop{\textbf{Spec}} O_\mathcal{X}}$. In particular, we get a morphism from the locally ringed topos $\mathop{\textbf{Spec}} O_\mathcal{X}$ to the ringed topos $\mathcal{X}$; diagrammatically: $$\begin{CD} \mathop{\textbf{Spec}} O_\mathcal{X} @>{\ulcorner O_{\mathop{\textbf{Spec}} O_\mathcal{X}} \urcorner}>> \mathbf{B} (\textbf{LCRing}) \\ @VVV \Rightarrow @VVV \\ \mathcal{X} @>>{\ulcorner O_\mathcal{X} \urcorner}> \mathbf{B} (\textbf{CRing}) \end{CD}$$ It remains to be shown this is universal.

By the yoga of classifying toposes again, we start with the universal case, i.e. the inclusion $\mathbf{B} (\textbf{LCRing}) \hookrightarrow \mathbf{B} (\textbf{CRing})$. For this, it is helpful to know that $\mathbf{B} (\textbf{CRingP})$ is the classifying topos for rings equipped with prime filters. Here, by filter of a ring $O$ I mean a subset $F \subseteq O$ such that:

• given elements $a_0, \ldots, a_{n-1}$ of $O$, $\{ a_0, \ldots, a_{n-1} \} \subseteq F$ if and only if $a_0 \cdots a_{n-1} \in F$.

For example, the set $O^\times$ of units of $O$ is a filter of $O$ – in fact, $O^\times$ is the smallest filter of $O$, since any filter contains $1$ and is closed under factors. A filter $F$ as above is prime if:

• if $a_0 + \cdots + a_{n-1} \in F$, then for some $i$, $a_i \in F$.

Classically, a subset is a prime filter if and only if its complement is a prime ideal, but prime filters are what matter constructively. Also, $O^\times$ is prime if and only if $O$ is local – indeed, this is the constructively correct definition of local ring – and, more generally, $O [F^{-1}]$ is a local ring if $F$ is a prime filter of $O$. The universal ring equipped with a prime filter in $\mathbf{B} (\textbf{CRingP})$ is the (non-local!) ring $(A, U) \mapsto A$ with the prime filter obtained by pulling back the canonical one from the local ring previously defined, i.e.: $$(A, U) \mapsto \{ a \in A : a \text{ becomes invertible in } O_{\operatorname{Spec} A} (U) \}$$ Thus, to factorise $\mathbf{B} (\textbf{LCRing}) \hookrightarrow \mathbf{B} (\textbf{CRing})$ it suffices to give a prime filter of the universal local ring – well, there is an obvious choice: the unit filter. Thus we get a geometric morphism $\mathbf{B} (\textbf{LCRing}) \to \mathbf{B} (\textbf{CRingP})$, and (with a bit of thought) we see that it is a right adjoint right inverse, i.e.:

• the composite $\mathbf{B} (\textbf{LCRing}) \to \mathbf{B} (\textbf{CRingP}) \to \mathbf{B} (\textbf{LCRing})$ is isomorphic to the identity,
• there is a 2-morphism from the identity to the composite $\mathbf{B} (\textbf{CRingP}) \to \mathbf{B} (\textbf{LCRing}) \to \mathbf{B} (\textbf{CRingP})$, and
• these satisfy the triangle identities.

Now we return to the original problem. Given a locally ringed topos $\mathcal{Y}$, a ringed topos $\mathcal{X}$, and a morphism $(f, f^\sharp) : (\mathcal{Y}, O_\mathcal{Y}) \to (\mathcal{X}, O_\mathcal{X})$ of ringed toposes, we may form the following pullback diagram in $\mathcal{Y}$: $$\begin{CD} F_f @>>> O_\mathcal{Y}^\times \\ @VVV @VVV \\ f^{-1} O_\mathcal{X} @>>{f^\sharp}> O_\mathcal{Y} \end{CD}$$ Since $O_\mathcal{Y}^\times$ is a prime filter of $O_\mathcal{Y}$, $F_f$ is a prime filter of $f^{-1} O_\mathcal{X}$, so we obtain a commutative square in the bicategory of toposes: $$\begin{CD} \mathcal{Y} @>{\ulcorner (f^{-1} O_\mathcal{X}, F_f) \urcorner}>> \mathbf{B} (\textbf{CRingP}) \\ @VVV @VVV \\ \mathcal{X} @>>{\ulcorner O_\mathcal{X} \urcorner}> \mathbf{B} (\textbf{CRing}) \end{CD}$$ By the universal property of pullbacks, we get a factorisation: $$\begin{CD} \mathcal{Y} @>{\ulcorner (f^{-1} O_\mathcal{X}, F_f) \urcorner}>> \mathbf{B} (\textbf{CRingP}) \\ @VVV @| \\ \mathop{\textbf{Spec}} O_\mathcal{X} @>>> \mathbf{B} (\textbf{CRingP}) \\ @VVV @VVV \\ \mathcal{X} @>>{\ulcorner O_\mathcal{X} \urcorner}> \mathbf{B} (\textbf{CRing}) \end{CD}$$ Postcomposing the top half with the 2-morphism corresponding to $f^\sharp$ and the geometric morphism $\mathbf{B} (\textbf{CRingP}) \to \mathbf{B} (\textbf{LCRing})$ gives us a diagram of the form below: $$\begin{CD} \mathcal{Y} @>{\ulcorner (O_\mathcal{Y}, O_\mathcal{Y}^\times) \urcorner}>> \mathbf{B} (\textbf{CRingP}) @>>> \mathbf{B} (\textbf{LCRing}) \\ @V{\tilde{f}}VV \Rightarrow @| @| \\ \mathop{\textbf{Spec}} O_\mathcal{X} @>>> \mathbf{B} (\textbf{CRingP}) @>>> \mathbf{B} (\textbf{LCRing}) \end{CD}$$ The composite of the top row is, of course, the classifying morphism $\ulcorner O_\mathcal{Y} \urcorner : \mathcal{Y} \to \mathbf{B} (\textbf{LCRing})$. The composite of the bottom row is the classifying morphism of the canonical local ring $O_{\mathop{\textbf{Spec}} O_\mathcal{X}}$ in $\mathop{\textbf{Spec}} O_\mathcal{X}$. Furthermore, the choice of $F_f$ and the definition of $\tilde{f}$ ensure that this corresponds to a morphism of locally ringed toposes: first, we have the following pullback square in $\mathcal{Y}$, $$\begin{CD} F_f @>>> \tilde{f}{}^{-1} O_{\mathop{\textbf{Spec}} O_\mathcal{X}}^\times \\ @VVV @VVV \\ f^{-1} O_\mathcal{X} @>>> \tilde{f}{}^{-1} O_{\mathop{\textbf{Spec}} O_\mathcal{X}} \end{CD}$$ and moreover we have $(f^{-1} O_\mathcal{X}) [F_f^{-1}] \cong \tilde{f}{}^{-1} O_{\mathop{\textbf{Spec}} O_\mathcal{X}}$, hence $$\begin{CD} \tilde{f}{}^{-1} O_{\mathop{\textbf{Spec}} O_\mathcal{X}}^\times @>>> O_\mathcal{Y}^\times \\ @VVV @VVV \\ \tilde{f}{}^{-1} O_{\mathop{\textbf{Spec}} O_\mathcal{X}} @>>{\tilde{f}{}^\sharp}> O_\mathcal{Y} \end{CD}$$ is also a pullback square in $\mathcal{Y}$.

We have now factorised the given morphism $(f, f^\sharp)$ as a morphism of locally ringed toposes followed by the morphism $(\mathop{\textbf{Spec}} O_\mathcal{X}, O_{\mathop{\textbf{Spec}} O_\mathcal{X}}) \to (\mathcal{X}, O_\mathcal{X})$, proving the existence part of the universality of the latter. Uniqueness remains to be shown but I think that should be straightforward given everything I have already said.

Answer 2

Sketch.

The construction is essentially an adaption of the functor $\mathrm{Spec}$ from ringed spaces to locally ringed spaces that you mention. The difference is that we don't have any points to work with.

Let $(X,\mathcal{O}_X)$ be a ringed locale. Consider the preorder of those $(U,f)$, where $U \in \mathrm{Open}(X)$ and $f \in \mathcal{O}_X(U)$. We define $(U,f) \leq (V,g)$ when $U \leq V$ and $g|_U \in \mathcal{O}_X(U)$ becomes invertible in $\mathcal{O}_X(U)[f^{-1}]$, i.e., it divides a power of $f$. So in that case we get a homomorphism $$\mathcal{O}_X(V)[g^{-1}] \to \mathcal{O}_X(U)[f^{-1}]. \tag{1}$$ Notice that finite meets exist: $$(U,f) \wedge (V,g) = (U \wedge V, f|_{U \wedge V} \cdot g|_{U \wedge V})$$ Consider the free partial order associated to this preorder. So in particular, we identify $(U,f)$ with $(U,f^n)$. Then, consider the frame freely generated by this partial order modulo the following relation:

• When $U = \bigvee_i U_i$ and $f \in \mathcal{O}_X(U)$, then $(U,f)$ is supposed to be equivalent to $\bigvee_i (U_i,f|_{U_i})$.

Edit. As Zhen Lin comments, some further relation is missing that used the additive structure.

Let $X_{\mathrm{loc}}$ be the locale whose frame of opens is this frame. Define a presheaf of commutative rings on $X_{\mathrm{loc}}$ by $$\mathcal{O}'_{X_{\mathrm{loc}}}(U,f) := \mathcal{O}_X(U)[f^{-1}],$$ using $(1)$ for restrictions, suitably extended to the whole frame of opens. Let $\mathcal{O}_{X_{\mathrm{loc}}}$ be the sheaf of commutative rings associated to $\mathcal{O}'_{X_{\mathrm{loc}}}$. Then $(X_{\mathrm{loc}},\mathcal{O}_{X_{\mathrm{loc}}})$ is locally ringed (!).

The map of frames $U \mapsto (U,1)$ induces a map of locales $\pi : X_{\mathrm{loc}} \to X$, and the natural homomorphisms $\mathcal{O}_X(U) \to \mathcal{O}'_{X_{\mathrm{loc}}}(U,1) \to \mathcal{O}_{X_{\mathrm{loc}}}(U,1)$ define a homomorphism of sheaves $\pi^\sharp : \mathcal{O}_X \to \pi_* \mathcal{O}_{X_{\mathrm{loc}}}$. Thus, $(\pi,\pi^\sharp) : (X_{\mathrm{loc}},\mathcal{O}_{X_{\mathrm{loc}}}) \to (X,\mathcal{O}_X)$ is a morphism of ringed locales.

Now one needs to verify that $(\pi,\pi^\sharp)$ is universal. Let me sketch the argument. Let $(\alpha,\alpha^\sharp) : (Y,\mathcal{O}_Y) \mapsto (X,\mathcal{O}_X)$ be a morphism, where $(Y,\mathcal{O}_Y)$ is a locally ringed locale. We extend the map of frames $\alpha^* : \mathrm{Open}(X) \to \mathrm{Open}(Y)$ to $\mathrm{Open}(X_{\mathrm{loc}})$ as follows. If $U \in \mathrm{Open}(X)$ and $f \in \mathcal{O}_X(U)$, we define $\alpha^*(U,f)$ as the largest open in $Y$ which is contained in $\alpha^*(U)$ and on which $\alpha^{\sharp}(f)$ is invertible. The sheaf homomorphism is extended by using the universal property of localizations.