Analytic stacks

Section 21 Analytic stacks

In this section, we define analytic stacks and give some examples.

Reference.

This section is based on [8], Lecture 19. Remark 21.3.1 is taken from [8], Lecture 24.

Subsection 21.1 Definition of analytic stacks

Definition 21.1.1.

An analytic stack is an accessible (Remark 19.2.1) functor \(F\colon \AnRing \to \Set\) with the following property: for any hypercovering \(\AnSpec(A_\bullet) \to \AnSpec(A)\) by \(!\)-able maps for which \(\calD(\Mod_A) \cong \lim^! \calD(\Mod_{A_\bullet})\text{,}\) we have

\begin{equation*} F(A) \cong \lim X(A_\bullet)\text{.} \end{equation*}

Let \(\AnStack\) be the category of analytic stacks.

For example, any representable functor is an analytic stack (and by Yoneda this defines a full embedding \(\AnRing^{\op} \to \AnStack\)). With this, the accessibility condition ensures that any object is a small colimit of representable functors.

Remark 21.1.2.

This is not quite the definition of [8] because we are taking functors valued in sets, not something more simplicial. This difference is certainly relevant but will not appear in the examples we consider.

Example 21.1.3.

Adic spaces, solid adic spaces, and dagger spaces all give rise to analytic stacks; in particular, the last of these include complex analytic spaces and Berkovich spaces. However, it is not clear that these functors are fully faithful.

Example 21.1.4.

There is a functor \(\Sch \to \AnStack\) taking \(\Spec A\) to \(\AnSpec \underline{A}\) where the discrete condensed ring \(\underline{A}\) is interpreted as a discrete analytic ring. This functor is fully faithful by a theorem of Bhatt.

This construction extends further to algebraic stacks. For example, using Stone duality (Definition 19.3.3) we get a fully faithful functor \(\CSet \to \AnStack\text{.}\) This allows to give meaning to the assertion that the definition of a norm function on an analytic ring (Definition 18.2.2) corresponds to a morphism \(\AnSpec A \to [0,\infty]\) of analytic stacks; in this formulation, one can extend the definition immediately to arbitrary analytic stacks.

In addition, if we start with an analytic ring \(A\text{,}\) we can consider schemes over \(\Spec A^{\triangleright}\) and form the functor taking \(\Spec B\) to \(\AnSpec \underline{B}\) where \(\underline{B}\) is given the analytic ring structure induced from \(A\text{.}\) This again defines a functor to analytic stacks.

Subsection 21.2 Flavors of the Riemann sphere

To illustrate the richness of the category of analytic stacks, we indicate how some different flavors of the Riemann sphere interact within this category.

Example 21.2.1.

We can form a sequence of morphisms of analytic stacks

\begin{equation*} X_1 \to X_2 \to X_3 \to X_4 \to X_5 \to X_6 \end{equation*}

in which:

\(X_1\) is the topological manifold \(S^2\text{.}\) This corresponds to taking the ring \(\Cts(S^2, \CC)\) over \(\CC_{\liquid}\text{.}\)
\(X_2\) is the \(C^\infty\) manifold \(S^2\text{.}\) This again corresponds to take the ring of global sections.
\(X_3\) is the real analytic manifold \(S^2\text{.}\) This cannot be treated globally; instead it is created by glueing as in Section 17.
\(X_4\) is the complex analytic manifold \(S^2\text{.}\) This is again created by glueing.
\(X_5\) is the scheme \(\PP^1_\CC\text{.}\) This is again created by glueing.
\(X_6\) is the topological space \(S^2\) treated as a condensed set. As a reminder, this is created by covering \(S^2\) with a profinite set.

Theorem 17.5.3 implies that \(X_4 \to X_5\) is an isomorphism of analytic stacks! The other maps are not isomorphisms.

Subsection 21.3 A final remark

Remark 21.3.1.

One potential use of this setup is to construct the sort of virtual fundamental classes which are needed all over the place in enumerative geometry. These are needed to count intersections of two geometric objects in cases where the “expected dimension” of the intersection is 0 but things can sometimes behave more badly. This is fairly familiar in algebraic geometry, where various techniques exist to handle this (moving lemmas, deformation to the normal cone) but is much more problematic in less rigid geometries, such as symplectic geometry (where such enumerative questions are fundamental in mirror symmetry).

A model case of this can be seen by considering \(R = \Cts(\RR, \RR)\) as a condensed ring. It can be shown that for \(f \in R\text{,}\) the isomorphism class of the quotient \(R/(f)\) determines \(f\) up to a unit; in particular this can be used to detect whether \(f\) has a sign crossing at one of its zeroes, even when the Taylor series of \(f\) is identically zero.

Subsection 21.4 Material to be removed

Lemma 21.4.1.

Let \(A\) be a solid analytic ring. Then for any \(f \in A(*)\text{,}\) for \(g \in \{1, 1-f\}\text{,}\) for any \(M \in \Mod_A\text{,}\) the sequence

\begin{equation*} 0\to M \to M \otimes_A A^{\triangleright} \left\langle \tfrac{f}{g} \right\rangle \oplus M \otimes_A A^{\triangleright} \left\langle \tfrac{g}{f} \right\rangle \to M \otimes_A A^{\triangleright} \left\langle \tfrac{f}{g}, \tfrac{g}{f} \right\rangle \to 0 \end{equation*}

is exact.

Proof.

Since the map in Lemma 13.2.1 (taking \(R := A^{\triangleright}\)) is an isomorphism, it remains an isomorphism after tensoring with \(A\text{.}\) Meanwhile, by Corollary 13.1.5, the sequences used in the proof of Lemma 13.2.2 remain exact after tensoring with \(M\text{,}\) so we may emulate the argument therein.

Definition 21.4.2.

Let \(A \to B\) be a rational localization of solid analytic rings corresponding to parameters \(f_1,\dots,f_n,g\in A^{\triangleright}(*)\) which generate the unit ideal. We can then factor this map through \(A \to B_0\) where

\begin{equation*} B_0 = (B^{\triangleright}, \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A)\text{;} \end{equation*}

we call the map \(A \to B_0\) the loose rational localization defined by \(f_1,\dots,f_n,g\text{.}\)

Define the loose adic topology on \(\AnRing^{\op}\) as the Grothendieck topology generated by coverings by families of loose rational localizations \(A \to A_{i,0}\) arising from sequences \(f_1,\dots,f_n \in A^{\triangleright}(*)\) generating the unit ideal.

We start with the following generalization of Corollary 13.2.7.

Lemma 21.4.3.

For \(A\) a solid analytic ring, for \(f,g \in A^{\triangleright}(*)\) which generate the unit ideal, for any \(A \in \Mod_A\text{,}\) the sequence

\begin{equation*} 0 \to M \to M \otimes_A A^{\triangleright} \left\langle \frac{f}{g} \right\rangle \oplus M \otimes_A A^{\triangleright} \left\langle \frac{g}{f} \right\rangle \to M \otimes_A A^{\triangleright} \left\langle \frac{f}{g}, \frac{g}{f} \right\rangle \to 0 \end{equation*}

is exact.

Proof.

In the case where \(g=1\) or \(g=1-f\) this was already established in Lemma 21.4.1. By this plus Proposition 12.3.8, we may check the general case locally with respect to a refinement of the binary covering defined by \(f,g\text{.}\) In particular, we may assume without loss of generality that \(A = A^{\triangleright} \left\langle \frac{f}{g} \right\rangle\text{,}\) in which case also \(A^{\triangleright} \left\langle \frac{g}{f} \right\rangle = A^{\triangleright} \left\langle \frac{f}{g}, \frac{g}{f} \right\rangle\) and so the sequence in question is split exact.

Lemma 21.4.4.

For \(A\) a solid analytic ring, for \(f,g \in A^{\triangleright}(*)\) which generate the unit ideal, the map

\begin{equation*} \Mod_A \to \Mod_{A \left\langle f/g \right\rangle_0} \times_{\Mod_{A \left\langle f/g, g/f \right\rangle_0}} \Mod_{A \left\langle g/f \right\rangle_0} \end{equation*}

is fully faithful.

Proof.

This is immediate from Lemma 21.4.3.

Before continuing, it will be helpful to make a corresponding statement for derived categories.

Theorem 21.4.5.

The categories \(\Mod_A\) form a stack for the loose adic topology on \(\AnRing^{\op}\text{.}\)

Proof.

Usiny Proposition 12.3.8, this reduces to Lemma 21.4.7.

Remark 21.4.6.

The truth of Theorem 21.4.5 may seem surprising in light of the fact that tensoring with even a loose rational localizations is not an exact morphism (Example 9.1.6). The point is that loose rational localizations are at least universally injective (Corollary 13.1.5) and this is sufficient for descent of modules (compare [28], tag 08WE for the case of modules over discrete rings). However, the failure of base extension along a loose rational localization to be flat is a compelling argument to introduce derived categories at this point.

Lemma 21.4.7.

For \(A\) a sheafy solid analytic ring, for \(f,g \in A^{\triangleright}(*)\) which generate the unit ideal, the map

\begin{equation*} \Mod_A \to \Mod_{A \left\langle f/g \right\rangle_0} \times_{\Mod_{A \left\langle f/g, g/f \right\rangle_0}} \Mod_{A \left\langle g/f \right\rangle_0} \end{equation*}

is an equivalence of categories.

Proof.

By Lemma 21.4.4, the functor is fully faithful. To check essential surjectivity, we may apply Remark 14.1.2 to an element of the target category to obtain a complex concentrated in degrees 0 and 1. However, we can then apply Lemma 21.4.4 again to see that the cohomology in degree 1 vanishes.

Remark 21.4.8.

The issues raised in Remark 21.4.6 and Remark 14.1.3 become more acute when we switch from the loose adic topology back to the adic topology, where Lemma 21.4.7 no longer applies: the base extension along a rational localization includes an analytic completion step which is not exact. So the best we can hope to assert is that some sort of derived category of \(\Mod_A\) is a stack for the adic topology, but this is obstructed in the same manner as in Remark 14.1.3.

As a first step, we can at least formulate the analogue of Remark 14.1.2. A complete resolution will have to wait until we introduce the \(\infty\)-categorical analogue of the derived category of \(\Mod_A\text{.}\)

Lemma 21.4.9.

Let \(A\) be a solid analytic ring and choose \(f,g \in A^{\triangleright}(*)\) generating the unit ideal. Then the base extension functor

\begin{equation} D(\Mod_A) \to D(\Mod_{A \left\langle f/g \right\rangle}) \times_{D(\Mod_{A \left\langle f/g, g/f \right\rangle})} D(\Mod_{A \left\langle g/f \right\rangle})\tag{21.1} \end{equation}

is an equivalence of categories.

Proof.

The proof of Remark 14.1.2 carries over.

To see that \(\underline{\ZZ[q]} \to P\) is flat, we may argue as in Lemma 13.1.3: form the exact sequence

\begin{equation*} 0 \to \underline{\ZZ[q]} \otimes P \stackrel{\times (q-[1])}{\to} \underline{\ZZ[q]} \otimes P \to P \to 0\text{,} \end{equation*}

then observe that for any \(M \in \Mod_{\underline{\ZZ[q]}}\text{,}\) \(M \otimes_{\underline{\ZZ[q]}} \underline{\ZZ[q]} \otimes P \cong M \otimes P\) injects into \(M \otimes (P \oplus \prod_{n \lt 0} \ZZ[n])\) on which multiplication by \(q-[1] = [1](1 - q[-1])\) is an isomorphism.

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