Glueing and GAGA for \PP^1 nonarchimedean case

Section 16 Glueing and GAGA for \(\PP^1\text{:}\) nonarchimedean case

In this section, we study the GAGA phenomenon for solid analytic rings, focusing on the case of \(\PP^1\) where we can state all the results in language that we (mostly) already have. We will return to the general case once we have the language of analytic stacks available to formulate the results more easily.

Reference.

This section is based on [8], Lecture 15.

Subsection 16.1 Analytification

We recall some ways in which schemes interact with other geometric objects. This is a behavior we wish to maintain in the category of algebraic stacks.

Remark 16.1.1.

In the category \(\LRS\) of locally ringed spaces, 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 \(\LRS \to \Ring^{\op}\text{.}\)

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. We can carry this further by replacing \(\LRS\) with our new category \(\LCRS\) of locally condensed ringed spaces. In that category we have the solid adic spaces introduced in Section 13.

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!

Definition 16.1.2.

For \(X\) a scheme locally of finite type over \(\CC\text{,}\) an analytification of \(X\) is a complex analytic variety \(X^{\an}\) equipped with a morphism \(X^{\an} \to X\) of locally ringed spaces, satisfying the following universal property: for any complex analytic variety \(Y\text{,}\) any morphism \(Y \to X\) of locally ringed spaces factors uniquely through \(X^{\an}\text{.}\)

Proposition 16.1.3.

Every scheme locally of finite type \(X\) over \(\CC\) admits an analytification \(X^{\an}\text{.}\) Moreover, the image of \(X^{\an} \to X\) consists of precisely the closed points of \(X\text{.}\)

Proof.

For \(X = \Spec \CC[T_1,\dots,T_n]\text{,}\) the variety \(X^{\an} = \CC^n\) and the mapping \(X^{\an} \to X\) corresponding to the identification of \(T_1,\dots,T_n\) with the coordinate functions on \(\CC^n\) does the trick. We may then formally extend the conclusion to any closed subscheme of \(X\text{,}\) and then to any scheme locally of finite type over \(\CC\text{.}\)

Theorem 16.1.4. 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.

We will recover this statement later (Theorem 17.5.5).

Remark 16.1.5.

One can similarly study analytification, and prove a GAGA theorem, for schemes over a nonarchimedean field \(K\) using any of rigid analytic spaces, Berkovich spaces, or adic spaces.

In the context of analytic stacks, we will upgrade Theorem 16.1.4 to a comparison of quasicoherent sheaves. This will amount to the statement that \(\PP^1_\CC\) and its analytification become isomorphic when viewed as analytic stacks.

Remark 16.1.6.

In nonarchimedean geometry, one can build the analytic analogue of \(\PP^1\) by glueing together two closed unit discs along their boundary. This works less well in complex geometry, but already in \(p\)-adic geometry there is a reason why one might want to change course.

This reason comes from de Rham cohomology: the cokernel of the derivation \(\frac{d}{dT}\) on the ring \(\QQ_p \langle T \rangle\) is infinite-dimensional! For example, \(\sum_{n=0}^\infty p^n T^{p^n}\) represents a nonzero element of the cokernel.

By contrast, if one replaces the Tate algebra \(\QQ_p \langle T \rangle\) with the dagger algebra \(\QQ_p \langle T \rangle^\dagger\text{,}\) meaning the subring consisting of those power series which converge on some disc of radius greater than 1 (depending on the particular series), then the cokernel of \(\frac{d}{dT}\) becomes zero!

Of course one can achieve the same effect simply by working with open discs instead of closed discs. The contrast between these approaches amounts to the distinction between dagger spaces and Berkovich spaces.

Subsection 16.2 Quasicoherent sheaves on solid \(\PP^1\)

We look more closely at the solid analytification of \(\PP^1\text{,}\) where we can build on our previous analysis. (One can treat \(\PP^n\) using similar techniques but slightly more notation.)

Definition 16.2.1.

For any analytic ring \(A\text{,}\) equip \(A[T], A[T^{-1}], A[T^{\pm}]\) with the induced analytic ring structures; that is,

\begin{equation*} \Mod_{A[T]} = \Mod_{A^{\triangleright}[T]} \times_{\Mod_{A^{\triangleright}}} \Mod_A \end{equation*}

and so on. Define the object \(\PP^1_A \in \LCRS\) by glueing \(\AnSpec A[T]\) and \(\AnSpec A[T^{-1}]\) along \(\AnSpec A[T^{\pm}]\text{.}\) This coincides with the fiber product \(\AnSpec A \times_{\Spec \ZZ} \PP^1_{\ZZ}\) in \(\LCRS\text{.}\)

Definition 16.2.2.

For \(A\) a solid analytic ring, define the object \(\PP^1_{A \solid} \in \LCRS\) by glueing \(\AnSpec A\langle T \rangle\) and \(\AnSpec A\langle T^{-1} \rangle\) along \(\AnSpec A\langle T^{\pm} \rangle\text{.}\) Here \(A \langle T \rangle\) denotes the analytic ring

\begin{equation*} (A^{\triangleright} \langle T \rangle, \Mod_{A^{\triangleright} \langle T \rangle \liquid T} \times_{\Mod_{A^{\triangleright}}} \Mod_A) \end{equation*}

and so on. Using this, we obtain a natural morphism \(\PP^1_{A\solid} \to \PP^1_{A}\text{.}\)

Lemma 16.2.3.

For any solid analytic ring \(A\text{,}\) the morphisms

\begin{equation*} A[T] \to A \langle T \rangle, \qquad A[T] \to A \langle T^{-1} \rangle [T] \end{equation*}

form an idempotent covering.

Proof.

We reduce to the case \(A = \underline{\ZZ}_{\solid}\text{.}\) In this case \(A\) is sheafy, so we may apply Example 15.3.6 to see that the two maps are idempotent, and we calculate directly that

\begin{equation*} 0 \to \underline{\ZZ}[T] \to \underline{\ZZ} \langle T \rangle \oplus \underline{\ZZ} \langle T^{-1} \rangle [T] \to \underline{\ZZ} \langle T^{\pm} \rangle \to 0 \end{equation*}

is split exact by splitting

\begin{equation*} \underline{\ZZ} \langle T^{-1} \rangle [T] = \underline{\ZZ} [T] \oplus T^{-1} \underline{\ZZ} \langle T^{-1} \rangle\text{.} \end{equation*}

Theorem 16.2.4.

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

Proof.

By Lemma 16.2.3, pulling back the given covering of \(\PP^1_A\) yields idempotent coverings of the constituents of \(\PP^1_{A \solid}\text{,}\) to which we may apply Theorem 15.3.12. This yields the first assertion; the second then follows formally as in Remark 15.1.1.

Remark 16.2.5.

We will interpret Theorem 16.2.4 later as the statement that \(\PP^1_{A \solid} \to \PP^1_A\) is an isomorphism of analytic stacks.

To recover more familiar forms of GAGA, we must bring in the scheme \(\PP^1_{A^{\triangleright}(*)}\) which receives a canonical map from \(\PP^1_A\text{.}\)

Theorem 16.2.6.

For \(A\) a sheafy solid 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.

By Theorem 15.2.5, pullback of pseudocoherent complexes from \(\Spec A[T]\) to \(\AnSpec A[T]\text{,}\) from \(\Spec A[T^{-1}]\) to \(\AnSpec A[T^{-1}]\text{,}\) and from \(\Spec A[T^{\pm}]\) to \(\AnSpec A[T^{\pm}]\) yields equivalences of categories. Glueing these statements yields the first assertion. We deduce the assertions about perfect complexes and vector bundles similarly by applying Theorem 15.2.7, Corollary 15.2.8, and Corollary 15.2.9.

In the case where \(A\) is noetherian, coherent sheaves on \(\PP^1_A\) can be identified with pseudocoherent complexes concentrated in amplitudes \((-\infty, 0]\) with vanishing cohomology in degrees \(\lt 0\text{.}\) We may thus reduce to the previous paragraph.

Corollary 16.2.7.

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

Proof.

The point is that every quasicoherent sheaf on \(\PP^1_{A^{\triangleright}(*)}\) is ind-coherent. We can thus deduce the claim from Theorem 16.2.6.

Remark 16.2.8.

As per Remark 14.4.2, one can remove the sheafiness hypothesis in the preceding results by working in a context where one can form derived rational localizations.

We can now recover a more standard formulation of GAGA over a nonarchimedean field.

Definition 16.2.9.

Let \(K\) be a nonarchimedean field. For \(X\) a scheme locally of finite type over \(K\text{,}\) an analytification of \(X\) is an adic space \(X^{\an}\) over \(K\) equipped with a morphism \(X^{\an} \to X\) of locally ringed spaces over \(K\text{,}\) satisfying the following universal property: for any adic space \(Y\) over \(K\text{,}\) any morphism \(Y \to X\) of locally ringed spaces factors uniquely through \(X^{\an}\text{.}\)

Proposition 16.2.10.

Every scheme locally of finite type \(X\) over \(K\) admits an analytification \(X^{\an}\text{.}\)

Proof.

Again, it suffices to check that for the affine space \(X = \Spec K[T_1,\dots,T_n]\text{,}\) the ascending union of all closed polydiscs serves as an analytification; note that the resulting Huber rings are strongly noetherian and hence sheafy (Remark 13.3.5), and so the resulting solid analytic rings are also sheafy (Proposition 13.3.2). We may then formally extend the conclusion to any closed subscheme of \(X\text{,}\) and then to any scheme locally of finite type over \(\CC\text{.}\)

Theorem 16.2.11. Nonarchimedean GAGA.

Let \(K\) be a nonarchimedean field. For \(X\) a proper scheme over \(K\text{,}\) 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.

For \(X = \PP^1_K \times_K \cdots \times_K \PP^1_K\text{,}\) this follows from Theorem 16.2.4 and Theorem 16.2.6 modulo one point: we need to know that the local rings of \(X^{\an}\) are coherent, in order to resolve coherent sheaves on \(X^{\an}\) into pseudocoherent complexes. Since \(X^{\an}\) is a noetherian adic space this is okay.

We can then deduce the case \(X = \PP^n_K\) using a Segre embedding (or by imitating the proof of Theorem 17.5.3); deduce the case where \(X\) is projective over \(K\) by pushing forward along the closed immersion; and deduce the case where \(X\) is proper over \(K\) using Chow’s lemma.

Remark 16.2.12.

One can similarly recover relative nonarchimedean GAGA (over some an affinoid ring), formal GAGA (comparing a scheme to its formal completion), and so on.

Subsection 16.3 Discrete \(\PP^1\)

We next return to the case of a discrete ring.

Lemma 16.3.1. Discrete Schwartz lemma.

Let \(A\) be a discrete condensed 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 \(A^{n}[-1] \to A^{n}[0]\text{.}\)

Proof.

We have a canonical identification \(A[\underline{S}]^\vee \cong \underline{\Cts(S, A})\text{;}\) in particular, any morphism \(A \to A[\underline{S}]^\vee \otimes A[\underline{S}]\) factors through \(A \to A[\underline{S_i}]^\vee \otimes A[\underline{S}]\) for some finite quotient \(S_i\) of \(S\text{.}\) We then do not change \(\cone(f-1)\) if we replace \(S\) with \(S_i\) by projecting along \(A[\underline{S}] \to A[\underline{S_i}]\text{:}\) more precisely, \(f\) acts on the exact sequence

\begin{equation*} 0 \to \ker(A[\underline{S}] \to A[\underline{S_i}]) \to A[\underline{S}] \to A[\underline{S_i}] \to 0 \end{equation*}

and the action of \(f\) on the left is zero, so the induced action of \(f-1\) on the left is invertible.

Theorem 16.3.2.

For \(A\) a discrete ring, pullback along \(\PP^1_{\underline{A}} \to \PP^1_A\) induces equivalences of categories (or stable \(\infty\)-categories) as follows:

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

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

Proof.

Using Lemma 16.3.1, we obtain an analogue of Theorem 15.2.5; we then argue as in Theorem 16.2.6.

Corollary 16.3.3.

For \(A\) a discrete 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.

Prev Top Next