In 1956 Jean-Pierre Serre published an important paper in Annales de l'institut Fourier called Géométrie algébrique et géométrie analytique, or GAGA. Using results from this other seminal paper Faisceaux algébriques cohérents (English: Coherent algebraic sheaves) or FAC, from 1955, he shows there is a really strong (and unexpected?) link between the analytic and algebraic world.

Ever since I learnt about this paper I wanted to do something with it, to pay tribute somehow. And in the meantime I could solidify my knowledge about it, because there are of course important assumptions and strong results underlying GAGA. I also wanted to know what does not hold, as of course the analytic and algebraic world aren't equivalent. I'm not sure whether the approach of summarizing GAGA in $24+\epsilon$ tweets (corresponding to the 24 paragraphs) in the paper is any good, so this is an experiment. A similar idea can be found in the MathOverflow thread Proof synopsis collection.

The $+\epsilon$ is to accomodate the fact that certain paragraphs are just definitions and don't contain necessarily mentionable ideas, and some paragraphs contain several. Being the good analyst I am by daily training (ahem) I can even quantify $\epsilon$ as 8. Remark that I count LaTeX symbols as 1 character, not by their macro length, as I could do everything in Unicode (but I don't like the results this gives). So the next section is both a table of contents of GAGA (which is non-existent in the real paper) and a listing of things I find especially interesting or important.

Recall before we start that a coherent sheaf $\mathcal{F}$ on a ringed space $X$ (either analytic or algebraic of nature) is a $\mathcal{O}_X$-module of finite type (i.e. there exists an open neighbourhood for each point such that the restriction of the sheaf is generated by a finite number of sections) and for every morphism $\mathcal{O}_X^n|_U\to\mathcal{F}|_U$ the kernel is of finite type). In the noetherian algebraic case this reduces to the adagium "looks locally like a finitely generated module", but in the analytic situation this notion is something more subtle (as far as I can tell). And of course the general ringed space situation is subtle, that's why my definition was wrong at first.

GAGA in $24+\epsilon$ tweets

§ 1. — Espaces analytiques.

1. Sous-ensembles analytiques de l'espace affine.

2. La notion d'espace analytique.

  1. an analytic space is locally an analytic subset with sheaf of holomorphic functions and separated as a topological space

3. Faisceaux analytiques.

  1. by Oka's coherence theorem the sheaf of holomorphic functions is coherent

4. Voisinage d'un point dans un espace analytique.

  1. an analytic variety is locally isomorphic to $\mathbb{C}^n$
  2. the local rings for an analytic variety are finite extensions of the (noetherian) rings of convergent power series in $\dim X$ variables

§ 2. — Espace analytique associé à une variété algébrique.

5. Définition de l'espace analytique associé à une variété algébrique.

6. Relations entre l'anneau local d'un point et l'anneau des functions holomorphes en ce point.

  1. every regular function being holomorphic $\mathcal{O}_x\hookrightarrow\mathcal{H}_x$ extends to isomorphism $\widehat{\mathcal{O}_x}\cong\widehat{\mathcal{H}_x}$
  2. $\mathcal{H}_x$ is flat over $\mathcal{O}_x$ by the previous tweet and tweet 32, and of same dimension

7. Relations entre la topologie usuelle et la topologie de Zariski d'une variété algébrique.

  1. an algebraic variety is complete if and only if it is compact
  2. Zariski and usual closure of the image of a regular function are the same

8. Un critère de régularité.

  1. if the graph of a holomorphic function is an algebraic variety the map is regular

§ 3. — Correspondance entre faisceaux algébriques et faisceaux analytiques cohérents.

9. Faisceau analytique associé à un faisceau algébrique.

  1. associated analytic sheaf is inverse image $\mathcal{F}^{\mathrm{h}}=f^*\mathcal{F}=f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}}\mathcal{H}$ for $f\colon X^{\mathrm{h}}\to X$ continuous
  2. by flatness (tweet 6) the functor $f^*$ is exact, and it sends an coherent algebraic sheaf to a coherent analytic sheaf because of this

10. Prolongement d'un faisceau.

  1. extension by zero is compatible with $f^*$ by associativity of the tensor product in the (non-zero) local rings

11. Homomorphismes induits sur la cohomologie.

  1. using Čech cohomology we get a natural morphism from cohomology in the algebraic context to cohomology in the analytic context

12. Variétés projectives. Énoncé des théorèmes.

  1. from now on we consider projective varieties (to reduce everything to $\mathbb{P}^n(\mathbb{C})$, not every algebraic variety is projective!
  2. Théorème 1. for an coherent algebraic sheaf $\mathcal{F}$ we get $\mathrm{H}^q(X,\mathcal{F})\cong\mathrm{H}^q(X^{\mathrm{h}},\mathcal{F}^{\mathrm{h}})$
  3. Théorème 2. the functor $f^*$ between categories of coherent sheaves is fully faithful
  4. Théorème 3. the functor $f^*$ between categories of coherent sheaves is essentially surjective

13. Démonstration du théorème 1.

  1. vanishing of structure sheaf by FAC or $(0,q)$ by Dolbeault, then a décalage argument with induction on the dimension for twists
  2. to conclude write as exact sequence with direct sum of twists, then two applications of weak five lemma

14. Démonstration du théorème 2.

  1. sheaf Hom, compatibility in stalks yields isomorphism by flatness of local rings
  2. $\mathrm{H}^0(X,\mathcal{H}\mathrm{om}(\mathcal{F},\mathcal{G}))\cong\mathrm{H}^0(X,\mathcal{H}\mathrm{om}(\mathcal{F},\mathcal{G})^\mathrm{h}$ by tweet 17, $\mathrm{H}^0(X,\mathcal{H}\mathrm{om}(\mathcal{F},\mathcal{G})^\mathrm{h})\cong\mathrm{H}^0(X,\mathcal{H}\mathrm{om}(\mathcal{F}^\mathrm{h},\mathcal{G}^\mathrm{h}))$ by tweet 17

15. Démonstration du théorème 3. Préliminaires.

  1. uniqueness by tweet 16, existence by tweet 12 and compatibility with ideal sheaves

16. Démonstration du théorème 3. Les faisceaux $\mathcal{M}(n)$.

  1. by Cartan A we get generated by global sections, by Cartan B we get décalage
  2. consider kernel of right exact sequence after tensoring with an analytic coherent sheaf, use tweet 23 to apply Nakayama

17. Fin de la démonstration du théorème 3.

  1. write the analytic coherent sheaf as cokernel of twists (i.e. come from algebraic side) and then use tweet 16

§ 4. — Applications.

18. Caractère algébrique des nombres de Betti.

  1. for non-singular algebraic varieties Betti numbers are invariant under field automorphisms by induced semilinear isomorphism on the cohomology

19. Le théorème de Chow.

  1. closed analytic subset of $\mathbb{P}^n(\mathbb{C})$ is algebraic (tweet 17): compact analytic subset algebraic is algebraic, ...
  2. ..., holomorphic map from compact algebraic variety to algebraic variety is regular, at most one algebraic structure on compact analytic

20. Espaces fibrés algébriques et espaces fibrés analytiques.

Although this section is really interesting it would require statements that don't fit in 140 characters. Cheating once in tweets 27 and 28 is enough. And it would be bad for the value of $\epsilon$.

Annexe.

21. Modules plats.

  1. if $E$ $A$-module of finite type, $B$ flat over $A$ then $\mathrm{Hom}_A(E,F)\otimes_AB\cong\mathrm{Hom}_B(E\otimes_AB,F\otimes_AB)$ by finite presentation

22. Couples plats.

  1. the term couple plat is equivalent to the inclusion morphism being faithfully flat

23. Modules sur un anneau local.

  1. for modules of finite type over noetherian local rings completion is equal to tensor product with completion of the ring

24. Propriétés de platitude des anneaux locaux.

  1. completion of local ring is faithfully flat, isomorphism of completions descends to flat couple, compatible with ideals and sees dimensions

Remarks

The three big results from complex analysis we use are Oka's coherence theorem (tweet 2) and Cartan A and B (tweet 23). I've been interested in their proof, but I never actually tried to read them thoroughly (they look scary at first and second sight). I might do this at some point, and if I can write something reasonable about it I will.

In tweet 18 I mention the idea of a décalage argument. Just like dévissage this is a French term which nicely describes a recurring technique in algebraic geometry. Maybe I should at some point write a blog post about these, it is my impression that the only way of learning the existence of these terms is by reading French texts on algebraic geometry which is something not everyone does. And once you have a word for something it becomes easier to remember.

So the main conditions we have used are that things are compact and projective.

Slogan

As this post is mostly about slogans anyway, here's one for GAGA.

If $X$ is a projective algebraic variety then we have for $\mathcal{F}$ a coherent (algebraic) sheaf that the cohomology in the algebraic case is isomorphic to the cohomology of the associated analytic sheaf on the associated analytic variety. The categories of coherent sheaves on $X$ and $X^\mathrm{h}$ are equivalent.

I would like to thank Mauro Porta for the discussions on this subject and this post. All remaining errors or incongruities are entirely my responsibility.