Glueing and GAGA for \PP^1 archimedean case and beyond

Section 17 Glueing and GAGA for \(\PP^1\text{:}\) archimedean case and beyond

In this section, we continue to study the GAGA phenomenon for analytic rings, but now introducing new terminology to handle the archimedean case in parallel. In particular, we recover the usual statement of GAGA over the complex numbers, and much more in the process.

Reference.

This section is very loosely based on [8], Lecture 15. The definition of dagger spaces is original.

Subsection 17.1 Dagger localization

In preparation to work over \(\CC\text{,}\) we revisit our development of solid adic spaces and quasicoherent sheaves thereupon.

Definition 17.1.1.

An analytic ring \(A\) is Tate if \(A^{\gas}\) contains a topologically nilpotent unit \(q \in A(*)\text{;}\) remember that this is a condition on all objects of \(\Mod_A\) (namely that \(\Mod_A \subseteq \Mod_{A^{\triangleright} \liquid q}\)), not just the base ring \(A^{\triangleright}\text{.}\) By Corollary 10.4.7, it follows that \(A^{\gas}\) contains every topologically nilpotent unit \(q \in A(*)\text{.}\)

Remark 17.1.2.

We can put solid analytic rings and Tate analytic rings on a common footing by considering an analytic ring \(A\) such that \(A^{\gas}\) contains a (fixed) unit \(q \in A(*)\text{:}\) a solid analytic ring corresponds to the case \(q=1\) while a Tate analytic ring corresponds to the case where \(q\) is topologically nilpotent. By Lemma 10.4.4, this condition is invariant under replacing \(q\) with \(q^m\) for any positive integer \(m\text{.}\)

Remark 17.1.3.

By analogy with Remark 13.1.1, at certain points we will need to consider a condensed ring \(R\) which not only has the property that \(R \in \Mod_{R \liquid q}\) for some unit \(q \in R(*)\text{,}\) but also has the property that \(R(*)\) contains a topologically nilpotent unit \(r\) which is not necessarily equal to \(q\text{.}\) When this happens, by Corollary 10.4.7 we will also have \(R \in \Mod_{R \liquid r}\text{.}\)

We can force ourselves into this situation by replacing \(R\) with \(R \otimes_{\underline{\ZZ}[q^{\pm}] \liquid q} A^{\triangleright}\) where \(A^{\triangleright}\) is a copy of the ring described in Definition 10.6.2, but with the label \(q\) replaced by \(r\text{.}\)

Definition 17.1.4.

Define the dagger Tate algebra in the variables \(T_1,\dots,T_n\) over \(\underline{\ZZ}[q^{\pm}]\) by the formula

\begin{equation*} \underline{\ZZ}[q^{\pm}] \langle T_1, \dots, T_n \rangle^\dagger_q := \colim_m \frac{\underline{\ZZ}[q^{\pm}]\langle U_1,\dots,U_n \rangle[T_1,\dots,T_n]}{(T_1^m q - U_1, \dots, T_n^m q - U_m)} \end{equation*}

where the transition map from \(m\) to \(m+1\) acts as \(T_i \mapsto T_i\text{;}\) this gives a well-defined map because

\begin{equation} (T_i^m q)^{m+1} = q (T_i^{m+1} q)^m\tag{17.1} \end{equation}

and so \(U_i^{m+1}\) maps to \(q U_i^m\text{.}\)

For \(R\) a condensed ring and \(q \in R(*)\) a unit, set

\begin{equation*} R \langle T_1, \dots, T_n \rangle^\dagger_q := R \otimes_{\ZZ[q^{\pm}] \liquid q} \underline{\ZZ}[q^{\pm}] \langle T_1, \dots, T_n \rangle^\dagger_q\text{.} \end{equation*}

We have the following extension of Lemma 12.2.3 (which is the case \(q=1\)).

Lemma 17.1.5.

For \(R\) a condensed ring such that \(R \in \Mod_{R \liquid q}\) for some unit \(q \in R(*)\) and \(R(*)\) contains a topologically nilpotent unit \(r\) (not necessarily equal to \(q\)),

\begin{equation*} R \langle T_1, \dots, T_n \rangle^\dagger_q \cong \colim_m R[T_1,\dots,T_n]_{\liquid T_1^m q, \dots, T_n^m q}\text{.} \end{equation*}

Proof.

To simplify notation we restrict to the case \(n=1\text{,}\) where the claim is that

\begin{equation*} R \langle T \rangle^\dagger_q \cong \colim_m R[T]_{\liquid T^m q}\text{.} \end{equation*}

By hypothesis, \(R\) is an algebra over \(R = \underline{\ZZ((r))}[q^{\pm}]\text{;}\) by Lemma 12.2.3 we may equate \(\underline{\ZZ((r))}[q^{\pm}] \langle U \rangle\) with \(\underline{\ZZ((r))}[q^{\pm}][U]_{\liquid U}\text{.}\) This gives the desired comparisons.

Definition 17.1.6.

For \(A\) an analytic ring and \(q \in A^{\gas}\) a unit, define

\begin{equation} A^{\triangleright} \left \langle \frac{f_1,\dots,f_n}{g} \right\rangle^{\dagger}_q := \frac{A^{\triangleright} \left \langle T_1,\dots,T_n \right\rangle^\dagger_q}{(f_1 - g T_1, \dots, f_n - g T_n)}\text{.}\tag{17.2} \end{equation}

Set

\begin{equation*} S := \{ \tfrac{f_1^m}{g^m} q, \dots, \tfrac{f_n^m}{g^m} q \colon m=1,2,\dots\} \end{equation*}

and let \(A\left \langle \frac{f_1,\dots,f_n}{g} \right\rangle^{\dagger}_q\) denote the analytic ring

\begin{equation*} \left(A^{\triangleright} \left \langle \frac{f_1,\dots,f_n}{g} \right\rangle^{\dagger}_q, \Mod_{A^{\triangleright} \left \langle \frac{f_1,\dots,f_n}{g} \right\rangle^{\dagger}_q \liquid S} \times_{\Mod_{A^{\triangleright}}} \Mod_A \right)\text{.} \end{equation*}

When \(q\) is topologically nilpotent, the construction does not depend on \(q\) (by Lemma 17.1.5 and Corollary 10.4.7) and so we omit \(q\) from the notation.

When \(f_1,\dots,f_n\) generate the unit ideal, we call this construction the dagger localization of \(A\) with respect to \(f_1,\dots,f_n,g\text{.}\) By calculating as in Remark 15.3.7, we see that \(A \to A\left \langle \frac{f_1,\dots,f_n}{g} \right\rangle^{\dagger}_q\) is an idempotent morphism provided that the quotient in (17.2) is resolved by the corresponding Koszul complex.

Remark 17.1.7.

Replacing the Tate algebra \(R \langle T \rangle\) with the dagger Tate algebra \(R \langle T \rangle^\dagger_q\) is analogous to replacing the usual Tate algebra \(K \langle T \rangle\) over a nonarchimedean field, whose elements represent holomorphic functions on the closed unit disc, with the dagger Tate algebra \(K \langle T \rangle^\dagger\text{,}\) whose elements represent holomorphic functions on some disc of radius strictly greater than \(1\text{.}\) The latter is more natural from the point of view of archimedean analysis; see for instance the proof of Theorem 17.5.5.

One feature we can take from this analogy is that \(K \langle T \rangle^\dagger\) can be equally well defined using a colimit over closed discs or open discs. Correspondingly, when \(q\) is topologically nilpotent we can also define \(\underline{\ZZ}[q^{\pm}] \langle T \rangle^\dagger_q\) in terms of the sequence space \(P\) using the formula

\begin{equation*} \colim_m \frac{(\underline{\ZZ}[q^{\pm}] \otimes P)[T]}{(T^m q - [1])}\text{,} \end{equation*}

where the transition map from \(m\) to \(m+1\) again acts as \(T \mapsto T\) (and similarly with more variables). To compare the two constructions, we use the fact that

\begin{equation*} (T^m q)^n = (T^{m+1} q)^{\lfloor \frac{nm}{m+1} \rfloor} T^{nm-(m+1)\lfloor \frac{nm}{m+1}\rfloor} q^{n - \lfloor \frac{nm}{m+1}\rfloor} \end{equation*}

to produce a map

\begin{equation*} \frac{(\underline{\ZZ}[q^{\pm}] \otimes P)[T]}{(T^m q - [1])} \to \frac{\underline{\ZZ}[q^{\pm}]\langle U \rangle[T]}{(T^{m+1} q - U)}\text{.} \end{equation*}

Remark 17.1.8.

Suppose that \(A\) is an analytic ring and \(q \in A^{\gas}\) is a topologically nilpotent unit. Following Remark 17.1.7, we may see that for any \(M \in \Mod_A\text{,}\) for each positive integer \(m\text{,}\)

\begin{equation*} M \otimes_A A^{\triangleright}[T_1,\dots,T_n]_{\liquid T_1^m q, \dots, T_n^m q} \in \Mod_{A^{\triangleright}[T_1,\dots,T_n] \liquid T_1^{m+1} q, \dots, T_n^{m+1} q}\text{.} \end{equation*}

From this it follows that for any dagger localization \(A \to B\text{,}\)

\begin{equation*} \Mod_B = \Mod_{B^\triangleright} \times_{\Mod_{A^{\triangleright}}} \Mod_A\text{;} \end{equation*}

that is, in the context of dagger localizations of Tate analytic rings, there is no need to keep track of the analogue of the ring of integral elements in the definition of a Huber pair.

Subsection 17.2 Interlude on \(q\)-coalescence

We need some analogues of the results of Section 7 where solid condensed modules over \(\underline{\ZZ}\) are replaced by \(q\)-coalescent condensed modules over \(\underline{\ZZ}[q^{\pm}]\) (carrying the discrete topology).

Lemma 17.2.1.

The coordinate map

\begin{equation*} \underline{\ZZ}[q^{\pm}] \otimes P \to \prod_\NN \underline{\ZZ}[q^{\pm}] \end{equation*}

induces an isomorphism

\begin{equation} (\underline{\ZZ}[q^{\pm}] \otimes P)_{\liquid q} \cong \colim_{j \in \NN} \prod_{n \in \NN} \bigoplus_{m=-j}^j \underline{\ZZ} q^{m+n}\tag{17.3} \end{equation}

including at the derived level over \(\underline{\ZZ}[q^{\pm}]\text{.}\)

Proof.

The map \(h\colon \underline{\ZZ}[q^{\pm}] \otimes P \to \underline{\ZZ}[q^{\pm}] \otimes P\) acting by \([n] \mapsto q^n[n]\) is an isomorphism and satisfies \(h\circ \Delta_q = \Delta_1 \circ h\text{.}\) It is thus sufficient to give an isomorphism

\begin{equation*} (\underline{\ZZ}[q^{\pm}] \otimes P)_{\liquid 1} \cong \colim_{j \in \NN} \prod_{n \in \NN} \bigoplus_{m=-j}^j \underline{\ZZ} q^m = \bigoplus_{m \in \ZZ} \left(\prod_{n \in \NN} \underline{\ZZ}\right) q^m \end{equation*}

at the derived level. Since \(\underline{\ZZ}[q^{\pm}]\) is projective in \(\CAb\text{,}\) computing the derived \(1\)-coalescence is equivalent over \(\underline{\ZZ}[q^{\pm}]\) and over \(\underline{\ZZ}\) (Remark 10.3.6). As the latter is just derived solidification of condensed abelian groups, we then deduce the claim directly from Corollary 7.2.12.

Lemma 17.2.2.

For any infinite \(S \in \Prof\text{,}\) for any morphism \(g\colon P \to \ZZ[\underline{S}]\) in \(\CAb\) as constructed in Definition 7.3.1, the map

\begin{equation*} 1\otimes g\colon \underline{\ZZ}[q^{\pm}] \otimes P \to \underline{\ZZ}[q^{\pm}] \otimes \ZZ[\underline{S}] \end{equation*}

induces an isomorphism of \(q\)-coalescences and of derived \(q\)-coalescences over \(\underline{\ZZ}[q^{\pm}]\text{.}\)

Proof.

The proof of Lemma 7.3.7 carries over.

Lemma 17.2.3.

For any \(S \in \Prof\text{,}\) the map

\begin{equation*} (\underline{\ZZ}[q^{\pm}][\underline{S}] \otimes P)_{\liquid q} \to \prod_n \underline{\ZZ}[q^{\pm}][\underline{S}]_{\liquid q} \end{equation*}

is an isomorphism; moreover, neither side changes when the \(q\)-coalescence is taken in the derived sense over \(\underline{\ZZ}[q^{\pm}]\text{.}\)

Proof.

If \(S\) is finite, then the claim follows immediately from Lemma 17.2.1. If \(S\) is infinite, we may apply Lemma 17.2.2 to rewrite the map in question as

\begin{equation*} (\underline{\ZZ}[q^{\pm}] \otimes P \otimes P)_{\liquid q} \to \prod_n (\underline{\ZZ}[q^{\pm}] \otimes P)_{\liquid q} \end{equation*}

and then apply Lemma 17.2.1 again.

Subsection 17.3 More on dagger localizations

Lemma 17.3.1.

For \(A\) a Tate analytic ring and \(M \in \Mod_A\text{,}\) the coordinate map

\begin{equation*} M \otimes^L_A A^{\triangleright} \langle T \rangle^\dagger \to \prod_n M \end{equation*}

is injective.

Proof.

We fix \(q \in A^{\gas}\) a topologically unit and formally reduce to the case where \(A = \underline{\ZZ}[q^{\pm}]_{\liquid q}\) (for the discrete topology on \(\underline{\ZZ}[q^{\pm}]\)). By virtue of Remark 17.1.7, it will suffice to check that for each positive integer \(m\text{,}\)

\begin{equation*} M \otimes^L_{\underline{\ZZ}[q^{\pm}] \liquid q} \frac{(\underline{\ZZ}[q^{\pm}] \otimes P)[T]}{(T^m q - [1])} \to \prod_n M \end{equation*}

is injective. This formally reduces to the case \(m=1\text{,}\) that is, showing that

\begin{equation*} M \otimes^L_{\underline{\ZZ}[q^{\pm}] \liquid q} (\underline{\ZZ}[q^{\pm}] \otimes P) \to \prod_n M \end{equation*}

is injective. We rewrite the left-hand side as

\begin{align*} (M \otimes^L_{\underline{\ZZ}[q^{\pm}]} (\underline{\ZZ}[q^{\pm}] \otimes P) )_{\liquid q} &= (M \otimes^L P)_{\liquid q} \\ &= (M \otimes P)_{\liquid q} \end{align*}

using Proposition 5.4.8 to replace the derived tensor product with the underived one; however, now we must treat the \(q\)-coalescence as being derived over \(\ZZ[q^{\pm}]\text{.}\)

We may further reduce to the case where \(M\) is the cokernel of some map \(\underline{\ZZ}[q^{\pm}][\underline{S}_1]_{\liquid q} \to \underline{\ZZ}[q^{\pm}][\underline{S}_0]_{\liquid q}\) with \(S_0, S_1 \in \Prof\) (here using underived \(q\)-coalescences). By combining Lemma 17.2.1 and Lemma 17.2.2, we deduce that:

\((\underline{\ZZ}[q^{\pm}][\underline{S_i}] \otimes P)_{\liquid q}\) is concentrated in degree \(0\text{;}\)
the coordinate map \(c_i\) from this object to \(\prod_n \underline{\ZZ}[q^{\pm}][\underline{S_i}]_{\liquid q}\) is injective;
if \(S_i\) is infinite, the image of the previous map is isomorphic to \(\colim_{j \in \NN} \prod_{n \in \NN} \bigoplus_{m=-j}^j \underline{\ZZ} q^{m+n}\text{.}\)

From this last point we may deduce that \(\coker(c_1) \to \coker(c_0)\) is injective; using the five lemma, we deduce the desired injectivity.

Lemma 17.3.2.

For \(A\) a Tate analytic ring, \(f \in A^{\triangleright}(*)\text{,}\) and \(M \in \Mod_A\text{,}\) multiplication by either \(f-T\) or \(1-fT\) is an injective map on \(M \otimes_A A^{\triangleright}[T]\) and \(M \otimes_A A^{\triangleright} \langle T \rangle^\dagger\text{.}\)

Proof.

Using Lemma 17.3.1 in place of Lemma 13.1.2, we argue as in the proofs of Lemma 13.1.3 (for multiplication by \(1-fT\)) and Lemma 13.1.4 (for multiplication by \(f-T\)).

Corollary 17.3.3.

For \(A\) a Tate analytic ring and \(f \in A^{\triangleright}(*)\text{,}\) the sequences

\begin{gather*} 0 \to A^{\triangleright}\langle T \rangle^\dagger \stackrel{\times (T-f)}{\to} A^{\triangleright}\langle T \rangle^\dagger \to A^{\triangleright} \langle f \rangle^\dagger \to 0\\ 0 \to A^{\triangleright}\langle T \rangle^\dagger \stackrel{\times (1-fT)}{\to} A^{\triangleright}\langle T \rangle^\dagger \to A^{\triangleright} \langle f^{-1} \rangle^\dagger \to 0 \end{gather*}

remain exact upon tensoring with any \(M \in \Mod_A\text{.}\) That is, the morphisms \(\times (T-f)\) and \(\times (1-fT)\) on \(A^{\triangleright}\langle T \rangle^\dagger_q\) in \(\Mod_A\) are universally injective.

Proof.

This is immediate from Lemma 17.3.2.

Lemma 17.3.4.

For \(A\) a Tate analytic ring, the map

\begin{equation*} A^{\triangleright} \langle T \rangle^\dagger \oplus T^{-1} A^{\triangleright} \langle T^{-1} \rangle^\dagger \to A^{\triangleright} \langle T^{\pm} \rangle^\dagger := A^{\triangleright} \langle T,U \rangle^\dagger / (TU-1) \end{equation*}

is an isomorphism.

Proof.

As in Lemma 13.2.1.

Lemma 17.3.5.

For \(A\) a Tate analytic ring and \(f,g \in A^{\triangleright}(*)\) with \(g \in \{1,1-f\}\text{,}\) the sequence

\begin{equation} 0 \to K \to A^{\triangleright}\left\langle \tfrac{f}{g} \right\rangle^\dagger \oplus A^{\triangleright} \left\langle \tfrac{g}{f} \right\rangle^\dagger \to A^{\triangleright}\left\langle \tfrac{f}{g}, \tfrac{g}{f} \right\rangle^\dagger \to 0\tag{17.4} \end{equation}

is exact for

\begin{equation*} K := \ker(\times (f-gT), A^{\triangleright} \langle T^{\pm} \rangle^\dagger\text{.} \end{equation*}

Proof.

We follow the proof of Lemma 13.2.2 using Corollary 17.3.3 in place of Lemma 13.1.3 and Lemma 13.1.4 and Lemma 17.3.4 in place of Lemma 13.2.1.

Definition 17.3.6.

We say that a Tate analytic ring \(A\) is dagger-sheafy if for every dagger localization \(A \to B\) and every \(f \in B^{\triangleright}\text{,}\) neither \(T-f\) nor \((1-f)T-f\) is a zero divisor on \(B^{\triangleright} \langle T^{\pm} \rangle^\dagger_q\text{.}\)

Theorem 17.3.7.

For \(A\) a dagger-sheafy Tate analytic ring and \(f_1,\dots,f_n \in A^{\triangleright}(*)\) generating the unit ideal, set

\begin{equation*} A_{i}^\dagger := A\left\langle \tfrac{f_1}{f_i},\dots,\tfrac{f_n}{f_i} \right\rangle^\dagger \quad (i=1,\dots,n)\text{.} \end{equation*}

Then the family \(\{A \to A_i^\dagger\}_i\) forms a covering for the idempotent topology on \(\AnRing^{\op}\text{.}\)

Proof.

We need to check the exactness of (15.3). Using Proposition 12.3.8, we may reduce to some applications of Lemma 17.3.5.

Theorem 17.3.8.

With notation as in Theorem 17.3.7, for \(S \subseteq \{1,\dots,n\}\) nonempty, let \(A_S^\dagger\) denote the tensor product (in \(\Mod_A\)) of \(A_i^\dagger\) over all \(i \in S\text{.}\) Then \(\calD(\Mod_A)\) is the limit of the diagram formed by the stable \(\infty\)-categories \(\calD(\Mod_{A_S^\dagger})\) over all \(S\text{.}\) Similarly, \(\Mod_A\) is the limit of the diagram formed by the categories \(\Mod_{A_S^{\dagger \triangleright}} \times_{\Mod_{A^\triangleright}} \Mod_A\) over all \(S\text{.}\)

Proof.

Theorem 15.3.12 Theorem 17.3.7

Subsection 17.4 Dagger spaces

We now restrict to the setting of a Tate analytic ring, where we can further introduce an analogue of solid adic spaces.

Definition 17.4.1.

For \(A\) a Tate analytic ring, define the valuative spectrum \(\Spv(A)\) as the set of valuations \(v\) on \(A^{\triangleright}(*)\) whose value on some (and hence any) topologically nilpotent unit \(q\) is cofinal. That is, for any \(a \in A^{\triangleright}(*)\text{,}\) either \(v(a) = 0\) or there exists some positive integer \(n\) such that \(v(q^n) \lt v(a)\text{.}\) This is again a spectral space ([16], Proposition 2.6(i)). As in Proposition 12.3.8, this topology is generated by compositions of standard binary coverings.

We define a presheaf \(\calO\) of condensed rings on \(\Spv(A)\) whose value on each rational subspace \(U(\tfrac{f_1,\dots,f_n}{g})\) (for \(f_1,\dots,f_n,g \in A^{\triangleright}(*)\) generating the unit ideal) the dagger localization \(A \left\langle \tfrac{f_1}{g}, \dots, \tfrac{f_n}{g} \right\rangle^{\dagger \triangleright}\text{.}\) Using Theorem 17.3.7, we see that \(\calO\) is a sheaf; we thus obtain an object of \(\LCRS\text{,}\) which we call the dagger spectrum of \(A\text{.}\)

A dagger space is an object of \(\LCRS\) locally of this form. For example, let \(\PP^1_{A \liquid}\) be the space obtained by glueing \(\AnSpec A \langle T \rangle^\dagger\) and \(\AnSpec A \langle T^{-1} \rangle^\dagger\) along \(\AnSpec A \langle T^{\pm} \rangle^\dagger\text{.}\)

Subsection 17.5 Dagger \(\PP^1\)

Theorem 17.5.1.

For \(A\) a dagger-sheafy Tate analytic ring, pullback along \(\PP^1_{A \liquid} \to \PP^1_A\) induces an equivalence of stable \(\infty\)-categories of derived categories of quasicoherent sheaves. Moreover, these equivalences are compatible with any combination of “bounded above”, “bounded below”, “compact”, “pseudocoherent”, “nuclear”, and “dualizable”.

Proof.

As in Theorem 16.2.4.

Lemma 17.5.2. Schwartz lemma for Tate analytic rings.

Let \(A\) be a Tate analytic ring and fix \(S \in \Prof\text{.}\) Then for any trace class morphism \(f\colon A[\underline{S}] \to A[\underline{S}]\text{,}\) \(\cone(f-1)\) is isomorphic in \(\calD(\Mod_{A})\) to a complex of the form \(A^{\triangleright n}[-1] \to A^{n}[0]\text{.}\)

Proof.

As in the proof of Lemma 15.2.3, we may lift to a setting where \(A^{\triangleright}\) is represented by a topological ring which is itself Tate. Now using the fact that \(A^{\circ \circ}\) is open in \(A^{\triangleright}(*)\text{,}\) we can imitate the proof of Lemma 15.2.3.

Theorem 17.5.3.

For \(A\) a dagger-sheafy Tate analytic ring, pullback along \(\PP^1_{A} \to \PP^1_{A^{\triangleright}(*)}\) induces equivalences of categories (or stable \(\infty\)-categories) as follows:

pseudocoherent complexes on \(\PP^1_{A^{\triangleright}(*)}\) to discrete pseudocoherent complexes on \(\PP^1_{A}\text{;}\)
perfect complexes on \(\PP^1_{A^{\triangleright}(*)}\) to discrete perfect complexes on \(\PP^1_{A}\text{;}\)
vector bundles on \(\PP^1_{A^{\triangleright}(*)}\) to vector bundles on \(\PP^1_{A}\text{.}\)

In addition, if \(A^{\triangleright}(*)\) is noetherian, we get an equivalence between coherent sheaves on \(\PP^1_{A^{\triangleright}(*)}\) and \(\PP^1_{A}\text{.}\)

Proof.

Using Lemma 17.5.2, we obtain an analogue of Theorem 15.2.5; we then argue as in Theorem 16.2.6.

Corollary 17.5.4.

For \(A\) a Tate analytic ring, pullback along \(\PP^1_{\underline{A}} \to \PP^1_A\) induces an equivalence of categories between the stable \(\infty\)-category of complexes of quasicoherent sheaves on \(\PP^1_A\) with amplitude in \((-\infty, 0]\) and the stable \(\infty\)-category of ind-pseudocoherent complexes on \(\PP^1_{\underline{A}}\) with amplitude in \((-\infty, 0]\text{.}\)

Proof.

As in Corollary 16.2.7.

Theorem 17.5.5. GAGA.

For \(X\) a proper scheme over \(\CC\text{,}\) pullback along \(X^{\an} \to X\) induces an exact equivalence of categories between coherent sheaves on \(X\) and \(X^{\an}\text{,}\) which moreover preserves sheaf cohomology.

Proof.

As in Theorem 16.2.11, we reduce to the case \(X = \PP^1_\CC \times_\CC \cdots \times_\CC \PP^1_\CC\text{,}\) which we can then deduce from Theorem 17.5.1 and Theorem 17.5.3 once we know that the local rings of \(X^{\an}\) are coherent. This follows from a theorem of Frisch stating that the ring of holomorphic functions on a closed unit polydisc is noetherian [9]. (The dagger-sheafy condition is automatic because the condensed rings in question are represented by topological rings of holomorphic functions on complex manifolds, and the vanishing of such functions is indeed a local condition.)

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