Given a complex variety $X,$ we have a functor $\mathrm{Mod}_{\mathscr{O}_{X}} \rightarrow \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}$ given by $\mathcal{F} \mapsto \mathcal{F}^{\mathrm{an}}.$ Recall that $\mathscr{F}^{\mathrm{an}} = u^{*}\mathscr{F}$ by definition, where $u : X^{\mathrm{an}} \rightarrow X$ is the unanalytification map. Using the adjoint bijection $$\mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(u^{*}\mathscr{F}, \mathscr{G}) \simeq \mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, u_{*}\mathscr{G}),$$ taking $\mathscr{G} = u^{*}\mathscr{F} = \mathscr{F}^{\mathrm{an}},$ we may obtain an $\mathscr{O}_{X}$-linear map $\mathscr{F} \rightarrow u_{*}\mathscr{F}^{\mathrm{an}}$ corresponding to the identity map of $\mathscr{F}^{\mathrm{an}}.$
What does the map look like? It is easier to use the more concrete definition $$u^{*}\mathscr{F} = u^{-1}\mathscr{F} \otimes_{u^{-1}\mathscr{O}_{X}} \mathscr{O}_{X^{\mathrm{an}}}.$$ Suppose that we are given a map $u^{*}\mathscr{F} \rightarrow \mathscr{G}.$ For any open $V \subset X^{\mathrm{an}},$ we have $$(u^{-1}\mathscr{F})(V) \rightarrow (u^{-1}\mathscr{F})(V) \otimes_{(u^{-1}\mathscr{O}_{X})(V)} \mathscr{O}_{X^{\mathrm{an}}}(V) \rightarrow \mathscr{G}(V),$$ so for any open $U \subset X,$ taking $V = U^{\mathrm{an}} = u^{-1}(U),$ we have $$(u^{-1}\mathscr{F})(u^{-1}(U)) \rightarrow \mathscr{G}(u^{-1}(U)).$$ Recall that $$(u^{-1}\mathscr{F})(V) = \mathrm{colim}_{W : u^{-1}(W) \subset V}\mathscr{F}(W),$$ where $W \subset X$ runs over all open subsets (with the specified condition), which gives us a map $\mathscr{F}(U) \rightarrow (u^{-1}\mathscr{F})(u^{-1}(U)).$ Hence, we get a map $$\mathscr{F}(U) \rightarrow (u^{-1}\mathscr{F})(u^{-1}(U)) \rightarrow \mathscr{G}(u^{-1}(U)) = (u_{*}\mathscr{G})(U).$$ This describes how to get $\mathscr{F} \rightarrow u_{*}\mathscr{G}.$
Explicit localization. Now, if $\mathscr{G} = u^{*}\mathscr{F}$ and we started with the identity map $u^{*}\mathscr{F} \rightarrow u^{*}\mathscr{F},$ then from the above description, the map $\mathscr{F} \rightarrow u_{*}u^{*}\mathscr{F}$ is given by $$\mathscr{F}(U) \rightarrow (u^{-1}\mathscr{F})(u^{-1}(U)),$$ followed by the tensor product and the sheafifications. Hence, if we compute the loacalization of $\mathscr{F} \rightarrow u_{*}u^{*}\mathscr{F}$ at $x \in X(\mathbb{C}),$ we get $$\mathscr{F}_{x} \rightarrow \mathscr{F}_{x} \otimes_{\mathscr{O}_{X,x}} \mathscr{O}_{X^{\mathrm{an}},x},$$ which is the same map as what we get by applying $\mathscr{F}_{x} \otimes_{\mathscr{O}_{X,x}} (-)$ to the local map $\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}.$
Next goal. We will show that $\mathscr{O}_{X^{\mathrm{an}},x}$ is Noetherian and the local map $$\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}$$ is flat. Due to the description above about the localization of $\mathscr{F} \rightarrow u_{*}\mathscr{F}^{\mathrm{an}},$ it would follow that this analytification map for $\mathscr{F}$ is exact.
Before we go into proving Noetherianity and flatness of the localization of the unanalytification map, we consider some statements from GAGA. For any ringed space $(X, \mathscr{O}_{X}),$ the category $\mathrm{Mod}_{\mathscr{O}_{X}}$ has enough injectives (Vakil Theorem 23.4.1), so the functor $\mathscr{F} \rightarrow u_{*}\mathscr{F}^{\mathrm{an}},$ induces $$H^{i}(X, \mathscr{F}) \rightarrow H^{i}(X, u_{*}\mathscr{F}^{\mathrm{an}}) = H^{i}(X^{\mathrm{an}}, \mathscr{F}^{\mathrm{an}}),$$ where the equality follows from the definition of the $i$-th right derived functor of global section functors of $u_{*}\mathscr{F}^{\mathrm{an}}$ and $\mathscr{F}^{\mathrm{an}}.$
1st GAGA. Let $X$ be a complete variety over $\mathbb{C}.$ Then the functor $$\mathrm{Mod}_{\mathscr{O}_{X}} \rightarrow \mathrm{Mod}_{\mathscr{O}_{X^{\mathrm{an}}}}$$ given by $\mathscr{F} \mapsto \mathscr{F}^{\mathrm{an}}$ restricts to a fully faithful functor $$\mathrm{Coh}(X) \rightarrow \mathrm{Coh}(X^{\mathrm{an}})$$ on coherent sheaves. Furthermore, for any coherent $\mathscr{F}$ and $\mathscr{G}$ on $X,$ the map $$\mathrm{Ext}^{i}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Ext}^{i}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}})$$ is an isomorphism.
Remark. We first need to explain what it means to be "coherent" means in $\mathrm{Mod}_{\mathscr{O}_{X}}$ for a general ringed space $(X, \mathscr{O}_{X}).$ We say an $\mathscr{O}_{X}$-module $\mathscr{F}$ is locally finitely generated if there is an open cover $X = \bigcup_{i \in I}U_{i}$ with some exact sequences $$\mathscr{O}_{U_{i}}^{n_{i}} \rightarrow \mathscr{F}|_{U_{i}} \rightarrow 0$$ for some $n_{i} \in \mathbb{Z}_{\geq 0}.$ A locally finitely generated $\mathscr{F}$ is coherent if
- $\mathscr{F}$ is locally finitely generated and
- for any open $U \subset X,$ the kernel of any morphism $\mathscr{O}_{U}^{\oplus n} \rightarrow \mathscr{F}|_{U}$ with any $n \in \mathbb{Z}_{\geq 0}$ is locally finitely generated.
When $X$ is a locally Noetherian scheme, the notion of coherence coincides with the notion of coherence in scheme theory (e.g., Chapter 5 Theorem 1.7 in Liu). Here $X$ being locally Noetherian makes the second condition above moot.
Why do we bother to think about more general definition of cohernece? There are two reasons I can think about.
- We do need the notion for the category of analytic spaces over $\mathbb{C}$.
- With this definition, coherent sheaves form an abelain category, so one can work with any scheme without any Noetherian conditions.
Personal remark. The above remarks are just cited without much of my soul involved.
We also need to understand how the maps between the Ext's are given. We first know the functor $\mathscr{F} \mapsto \mathscr{F}^{\mathrm{an}}$ gives $$\mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}}).$$ This is the $0$-th map $$\mathrm{Ext}^{0}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Ext}^{0}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}}).$$ The category $\mathrm{Mod}_{\mathscr{O}_{X}}$ has enough injectives (Vakil Theorem 23.4.1), so the $i$-th right derived functor $\mathrm{Ext}^{i}_{\mathscr{O}_{X}}(\mathscr{F}, -)$ of the left exact functor $$\mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, -) : \mathrm{Mod}_{\mathscr{O}_{X}} \rightarrow \textbf{Ab}$$ is a universal $\delta$-functor (Vakil Corollary 23.2.10). Hence, the morphism $$\mathrm{Hom}_{\mathscr{O}_{X}}(\mathscr{F}, -) \rightarrow \mathrm{Hom}_{\mathscr{O}_{X^{\mathrm{an}}}}(u_{*}\mathscr{F}^{\mathrm{an}}, u_{*}(-))$$ of functors giving the degree $0$ map must extend to higher degrees.
Still to do. I have not completed the description of the maps between Ext groups, so the 1st GAGA only makes sense to me for the case $i = 0$ yet. This may or may not work.
Personal remark. Quite a bit of the remarks in the preceding paragraphs are obtained by skimming various sources and conversations with some folks I know without detailed justifications, so I should perhaps come back to this at some point to check the details. Thanks to Nancy Wang, I realized that we cannot use free resolutions in this scenario. The reason is that the structure sheaf is rarely projective (e.g. the paragraph after 30.2.D in Vakil).
What I am told. I have consulted an expert, who suggested me to think about the description of $\mathrm{Ext}^{i}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G})$ as an equivalence class of exact sequences of the form: $$0 \rightarrow \mathscr{G} \rightarrow \mathscr{E}_{1} \rightarrow \cdots \rightarrow \mathscr{E}_{n} \rightarrow \mathscr{F} \rightarrow 0.$$ This is great because since the analytification functor on sheaves is exact, so we get $$0 \rightarrow \mathscr{G}^{\mathrm{an}} \rightarrow \mathscr{E}_{1}^{\mathrm{an}} \rightarrow \cdots \rightarrow \mathscr{E}_{n}^{\mathrm{an}} \rightarrow \mathscr{F}^{\mathrm{an}} \rightarrow 0.$$ This does give a map $\mathrm{Ext}^{i}_{\mathscr{O}_{X}}(\mathscr{F}, \mathscr{G}) \rightarrow \mathrm{Ext}^{i}_{\mathscr{O}_{X^{\mathrm{an}}}}(\mathscr{F}^{\mathrm{an}}, \mathscr{G}^{\mathrm{an}}).$ This description of elements of Ext seems quite general. (See Construction of Ext in abelian categories in this Wikipedia page.)
Remark. The future lectures will prove the 1st GAGA for the case when $X$ is projective over $\mathbb{C},$ and this case is due to Serre. The version for complete varieties over $\mathbb{C}$ is due to Grothendieck. There are (relative) versions for proper morphisms.
Given an analytic space $X,$ checking that $\mathscr{O}_{X}$ is coherent seems quite difficult. That is, how an earth would we check every map $\mathscr{O}_{X}^{n} \rightarrow \mathscr{O}_{X}$ with any $n \in \mathbb{Z}_{\geq 0}$ has a kernel that is locally finitely generated? Thankfully, this seems like a famous theorem:
Theorem (Oka). Given any analytic space $X,$ its structure sheaf $\mathscr{O}_{X}$ is coherent. In particular, vector bundles on $X$ give coherent $\mathscr{O}_{X}$-modules.
Personal remark. Where is a reference for Oka's theorem for analytic spaces? Maybe this "easily" follows from the Wikipedia version when $X$ is complex manifold?
Analytification of a coherent module is coherent. Let $X$ be a complex variety and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. Then $\mathscr{F}^{\mathrm{an}}$ is a coherent $\mathscr{O}_{X^{\mathrm{an}}}$-module.
Proof. Note that for any complex variety $X$ and a coherent $\mathscr{O}_{X}$-module $\mathscr{F},$ we can choose an exact sequence $$\mathscr{O}_{X}^{\oplus m} \rightarrow \mathscr{O}_{X}^{\oplus n} \rightarrow \mathscr{F} \rightarrow 0.$$ This induces $$\mathscr{O}_{X^{\mathrm{an}}}^{\oplus m} \rightarrow \mathscr{O}_{X^{\mathrm{an}}}^{\oplus n} \rightarrow \mathscr{F}^{\mathrm{an}} \rightarrow 0,$$ where the exactness can be checked at the stalks. Thus, by Oka's theorem, we see that $\mathscr{F}^{\mathrm{an}}$ is the cokernel of a morphism between coherent $\mathscr{O}_{X^{\mathrm{an}}}$-modules. Using the fact that coherent $\mathscr{O}_{X^{\mathrm{an}}}$-modules form an abelian category (which does not seem obvious), this proves that $\mathscr{F}^{\mathrm{an}}$ is coherent. $\Box$
Remark. The above proves a tiny bit of the 1st GAGA.
2nd GAGA. Let $X$ be a complete complex variety. The functor $\mathscr{F} \mapsto \mathscr{F}^{\mathrm{an}}$ gives an equivalence of the categories $$\mathrm{Coh}(X) \simeq \mathrm{Coh}(X^{\mathrm{an}}).$$ Here is an interesting consequence. We will prove that $\mathscr{O}_{X,x} \rightarrow \mathscr{O}_{X^{\mathrm{an}},x}$ is faithfully flat for every $x \in X(\mathbb{C}),$ so this tells us that the functor given for the 2nd GAGA restricts to the following equivalence of categories: $$\{\text{locally free coherent } \mathscr{O}_{X}\text{-modules}\} \simeq \{\text{locally free coherent } \mathscr{O}_{X^{\mathrm{an}}}\text{-modules}\}.$$ This equivalence evidently preserves the rank, so it gives an equivalence between vector bundles of fixed rank.
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