Little over a year ago Markus Reineke published a note on the arXiv titled Every projective variety is a quiver Grassmannian about how quiver Grassmannians can be used to describe every possible projective variety. Based on this note Lieven Le Bruyn wrote a blogpost Quiver Grassmannians can be anything, which explains the result using an easy to grasp example. An interesting discussion in the comments section ensued, which was lost in the move to Drupal but apparently restored yesterday (unrelated to my interest in this).

Yesterday I attended a talk by Sarah Scherotzke in which this result was mentioned, and this renewed my interest for it. To entertain myself tonight I implemented some code that determines the information as described in Lieven’s example for a general projective variety. The implementation is based on Reineke’s original proof, not on Michel Van den Bergh’s alternative proof. It is nothing extraordinary, it is really just a programming exercise.

The code is written in SAGE, and can be found in the gist pbelmans/5645902. Its output can be found at Pastebin. If you have any comments, feel free to post them.

Time for a little commercial break. A seminar in which I’ve been participating now has its own webpage, containing the set of notes people have written for this seminar. After April 26 there will be a short break, May 17 it will restart. That day I might or might not give a lecture on derived moduli stacks. Feel free to come if you are interested.

All remarks on the layout of the webpage (which I realize is rather bare, but I wasn’t feeling inspired and I’m a lousy webdesigner anyway) and the notes are appreciated.

I don’t know whether it’s common in other parts of the world, but in France one often (or rather: always) denotes an open or closed immersion by drawing a circle or a slash on the hooked arrow. A bad impression hacked together in plain LaTeX would be $\hookrightarrow\!\!\!\!\!\!\!\circ$ and $\hookrightarrow\!\!\!\!\!\!\!/$.

Now to define the six functors formalism we have a scheme $X$, a closed subscheme $Z$ and an open subscheme $U$ such that on the level of the underlying sets our big scheme is the disjoint union. In order to write this using tikz-cd, which is an excellent package to write commutative diagrams, a bit of effort is needed. We define the appropriate arrow styles by

\usetikzlibrary{decorations.markings}
\tikzset{
closed/.style = {decoration = {markings, mark = at position 0.5 with { \node[transform shape, xscale = .8, yscale=.4] {/}; } }, postaction = {decorate} },
open/.style = {decoration = {markings, mark = at position 0.5 with { \node[transform shape, scale = .7] {$\circ$}; } }, postaction = {decorate} }
}

and then we can produce the diagram by

\begin{tikzcd}
Z \arrow[hook, closed]{r}{i} & X \arrow[hookleftarrow, open]{r}{j} & U
\end{tikzcd}

This results in

Remark that we have to write the arrow from $U$ to $X$ in the opposite direction, because otherwise the hook is upside down. I haven’t found a better solution for this. And it seems that in the font I’m using (the Bera family with Charter from mathdesign for mathematics) the circle is slightly above the center of the line. If one goes this far in writing a silly diagram this should be fixed too, but unfortunately I haven’t found a (clean) solution for this yet. In the default font it looks better by the way.

It should be possible to do this in plain math mode too, but you’ll have to create long hooked arrows etc. This is easier and more consistent I think.

Tonight I was having dinner in the kitchen of my dorm and one of my neighbours who also studies math was with me. This led to a discussion on “the Rubik’s cube group“. Whether it’s any interesting I don’t know, but we asked ourselves:

• What if you take a sufficiently interesting subgroup (that is, not necessarily generated by a subset of standard moves) and only allow these moves to be used? They could be combinations of standard moves, hence you always do multiple moves in the same time. You’d need a computer program to play such a cube, but it might lead to interesting results. You’d have to train yourself to only consider moves which are part of the given subgroup. Would this be entertaining?
• Is the “minimal subgroup generated by moves to solve a state” an interesting invariant? This might be related to the diameter of the group. For instance I wonder whether there are two states for which the minimal number of moves satisfies a strict inequality, while the order of the minimal subgroups of moves satisfies the other.

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.

###### 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.

###### 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 $\mathrm{Hom}_A(E,F)\otimes_AB\cong\mathrm{H}om_B(E\otimes_AB,F\otimes_AB)$ 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.

It has been a while since I actively used TeX and encountered interesting issues, but as the work on my Master thesis has finally started I am again TeX’ing. It feels great!

The todonotes package is a great tool if you’re as disorganized as I am, whenever I have a random idea, or don’t feel like finishing something or don’t want to add the correct citation, etc. I add a \todo{} and postpone the work. Postponing is good.

And if you can postpone work in a fancy way it’s even better! So I’ve got 5 “categories” of notes, each with its own color and options. An example (click for pdf):

The interesting part is that I’m using the xparse package, to have optional commands. So I can choose between \addreference and \addreference{to that particular book for which I don't feel like creating a BibTeX entry right now}. It’s really easy, just put

\usepackage{xparse}


in your preamble when you have todonotes loaded. Adding the option textsize = small to your \usepackage command might be a good idea, and there is colorinlistoftodos if you want colored boxes in the list.