الهندسة الجبرية والهندسة التحليلية

1. Let ${\displaystyle (X,{\mathcal {O}}_{X})}$ be a scheme of finite type over ${\displaystyle \mathbb {C} }$. Then there is a topological space ${\displaystyle X^{\mathrm {an} }}$ that consists of the closed points of ${\displaystyle X}$ with a continuous inclusion map ${\displaystyle \lambda _{X}:X^{\mathrm {an} }\to X}$. The topology on ${\displaystyle X^{\mathrm {an} }}$ is called the "complex topology" (and is very different from the subspace topology).
2. Suppose ${\displaystyle \phi :X\to Y}$ is a morphism of schemes of locally finite type over ${\displaystyle \mathbb {C} }$. Then there exists a continuous map ${\displaystyle \phi ^{\mathrm {an} }:X^{\mathrm {an} }\to Y^{\mathrm {an} }}$ such that ${\displaystyle \lambda _{Y}\circ \phi ^{\mathrm {an} }=\phi \circ \lambda _{X}}$.
3. There is a sheaf ${\displaystyle {\mathcal {O}}_{X}^{\mathrm {an} }}$ on ${\displaystyle X^{\mathrm {an} }}$ such that ${\displaystyle (X^{\mathrm {an} },{\mathcal {O}}_{X}^{\mathrm {an} })}$ is a ringed space and ${\displaystyle \lambda _{X}:X^{\mathrm {an} }\to X}$ becomes a map of ringed spaces. The space ${\displaystyle (X^{\mathrm {an} },{\mathcal {O}}_{X}^{\mathrm {an} })}$ is called the "analytification" of ${\displaystyle (X,{\mathcal {O}}_{X})}$ and is an analytic space. For every ${\displaystyle \phi :X\to Y}$ the map ${\displaystyle \phi ^{\mathrm {an} }}$ defined above is a mapping of analytic spaces. Furthermore, the map ${\displaystyle \phi \mapsto \phi ^{\mathrm {an} }}$ maps open immersions into open immersions. If ${\displaystyle X=\operatorname {Spec} (\mathbb {C} [x_{1},\dots ,x_{n}])}$ then ${\displaystyle X^{\mathrm {an} }=\mathbb {C} ^{n}}$ and ${\displaystyle {\mathcal {O}} _ {X}^ {\mathrm {an} }(U)}$ for every polydisc ${\displaystyle U}$ is a suitable quotient of the space of holomorphic functions on ${\displaystyle U}$.
4. For every sheaf ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ (called algebraic sheaf) there is a sheaf ${\displaystyle {\mathcal {F}}^{\mathrm {an} }}$ on ${\displaystyle X^{\mathrm {an} }}$ (called analytic sheaf) and a map of sheaves of ${\displaystyle {\mathcal {O}}_{X}}$-modules ${\displaystyle \lambda _{X}^{*}:{\mathcal {F}}\rightarrow (\lambda _{X})_{*}{\mathcal {F}}^{\mathrm {an} }}$. The sheaf ${\displaystyle {\mathcal {F}}^{\mathrm {an} }}$ is defined as ${\displaystyle \lambda _{X}^{-1}{\mathcal {F}}\otimes _{\lambda _{X}^{-1}{\mathcal {O}}_{X}}{\mathcal {O}}_{X}^{\mathrm {an} }}$. The correspondence ${\displaystyle {\mathcal {F}}\mapsto {\mathcal {F}}^{\mathrm {an} }}$ defines an exact functor from the category of sheaves over ${\displaystyle (X,{\mathcal {O}}_{X})}$ to the category of sheaves of ${\displaystyle (X^{\mathrm {an} },{\mathcal {O}}_{X}^{\mathrm {an} })}$.The following two statements are the heart of Serre's GAGA theorem^[14][15] (as extended by Alexander Grothendieck, Amnon Neeman, and others).
5. If ${\displaystyle f:X\to Y}$ is an arbitrary morphism of schemes of finite type over ${\displaystyle \mathbb {C} }$ and ${\displaystyle {\mathcal {F}}}$ is coherent then the natural map ${\displaystyle (f_ {*}{\mathcal {F}})^{\mathrm {an} }\rightarrow f_{*}^{\mathrm {an} }{\mathcal {F}}^{\mathrm {an} }}$ is injective. If ${\displaystyle f}$ is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves ${\displaystyle (R^{i}f_{*}{\mathcal {F}})^{\mathrm {an} }\cong R^{i}f_{*}^{\mathrm {an} }{\mathcal {F}}^{\mathrm {an} }}$ in this case.^[16]
6. Now assume that ${\displaystyle X^{\mathrm {an} }}$ is Hausdorff and compact. If ${\displaystyle {\mathcal {F}},{\mathcal {G}}}$ are two coherent algebraic sheaves on ${\displaystyle (X,{\mathcal {O}}_{X})}$ and if ${\displaystyle f\colon {\mathcal {F}}^{\mathrm {an} }\rightarrow {\mathcal {G}}^{\mathrm {an} }}$ is a map of sheaves of ${\displaystyle {\mathcal {O}}_{X}^{\mathrm {an} }}$-modules then there exists a unique map of sheaves of ${\displaystyle {\mathcal {O}}_{X}}$-modules ${\displaystyle \varphi$ :{\mathcal {F}}\rightarrow {\mathcal {G}}} with ${\displaystyle f=\varphi ^{\mathrm {an} }}$. If ${\displaystyle {\mathcal {R}}}$ is a coherent analytic sheaf of ${\displaystyle {\mathcal {O}}_{X}^{\mathrm {an} }}$-modules over ${\displaystyle X^{\mathrm {an} }}$ then there exists a coherent algebraic sheaf ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-modules and an isomorphism ${\displaystyle {\mathcal {F}}^{\mathrm {an} }\cong {\mathcal {R}}}$.

In slightly lesser generality, the GAGA theorem asserts that the category of coherent algebraic sheaves on a complex projective variety ${\displaystyle X}$ and the category of coherent analytic sheaves on the corresponding analytic space ${\displaystyle X^{\mathrm {an} }}$ are equivalent. The analytic space ${\displaystyle X^{\mathrm {an} }}$ is obtained roughly by pulling back to ${\displaystyle X}$ the complex structure from ${\displaystyle \mathbb {C} ^{n}}$من خلال مخططات الإحداثيات. في الواقع، إن صياغة النظرية بهذه الطريقة أقرب في روحها إلى ورقة سير، بالنظر إلى أن لغة نظرية المخططات الكاملة التي يستخدمها البيان الرسمي أعلاه بشكل كبير لم تكن قد تم اختراعها بعد وقت نشر GAGA.