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Synthetic Algebraic Geometry in the Zariski-Topos

Stay updated on synthetic algebraic geometry by watching this repository, joining the next meeting or with the mailing list. Due to a bug in the mailinglist-service of chalmers, it does not work to just answer to the email-confirmation email - so use the link to confirm your email address instead. You know that you really signed up to the list, if you can login on the page linked above.

This is a latex documentation of our understanding of the synthetic theory of the Zariski-Topos and related ideas. The drafts below are currently built hourly - if you want to make sure you are viewing the latest built, CTRL+F5 should clear all caches in most browsers. There are currently the following preprints:

And the following drafts and notes:

There is a related formalization project. Here is an overview of the current ongoing work in SAG and related areas.

Questions

  • Is every étale proposition (formally étale and a scheme) an open proposition?
  • Is every étale scheme a sub-quotient of a finite set?
  • If $A$ is an étale $R$-algebra (finitely presented and the spectrum is étale), is it impossible to have an injective algebra map $R[X] \to A$?
  • Can every bundle (on $Sp A$) of strongly quasicoherent $R$-modules be recovered from its $A$-module of global sections?
  • Can we compute some interesting étale/fppf cohomology groups?
  • Is the intergral closure of $R$ in a finitely presented $R$-algebra $A$ finitely presented?

Answered Questions

  • Is the proposition "X is affine" not-not-stable, for X a scheme? (Then deformations ($D(1) \to \mathrm{Sch}$) of affine schemes would stay affine.)

    No: Let $X$ be an open proposition, then up to $\neg\neg$ it is $1$ or $\emptyset$, which are both affine, but we know that not all open propositions are affine.

  • Is $\mathrm{Spec} A$ quasi-complete ("compact") for $A$ a finite $R$-algebra (fin gen as $R$-module)?

    Yes: By the discussion in #5 and #6, $\mathrm{Spec} A$ is even projective, whenever $A$ is finitely generated as an $R$-module.

  • Can there be a flat-modality for $\mathbb{A}^1$-homotopy theory which has the same properties as the flat in real-cohesive HoTT?

    No: By the disucssion in #18, this should not be possible, because it would imply that the category of $\mathbb{A}^1$-local types is a topos, which is known to be false. There can still be a flat-modality with weaker properties, for example, the global section functor should generally induce such a modality.

  • For $f : A$, is $f$ not not zero iff $f$ becomes zero in $A \otimes R/\sqrt{0}$?

    No: for $r : R$, we have $r + (r^2)$ not not zero in $R/(r^2)$, but if it were always zero in $R/(r^2,\sqrt{0})$, then we would have a nilpotent polynomial $f : R[x]$ such that $x \in f + (x^2)$, which is false.

Learning material

There are some recordings of talks from the last workshop on synthetic algebraic geometry. And there is a hottest talk on the foundations article.

Building the drafts

We use latex now instead of xelatex, to be compatible with the arxiv. For each draft, a build command may be found at the start of main.tex.

Arxiv

To put one of the drafts on the arxiv, we have to

  • make sure there is a good abstract for the draft
  • make a temporary folder, e.g. synthetic-zariski/projective/tmp copy all tex-files there and run 
    ../../util/zar-rebase.sh ../../util/
  • run latexmk -pdf -pvc main.tex to produce the main.bbl and check if the draft builds.
  • put all the files into a .tar.gz, so everything can be uploaded in one step, e.g. 
    tar -czv -f DRAFT.tar.gz *.tex *.cls *.sty main.bbl

Watching this repo

... is a good idea since we started to use the issue-tracker

grafik

for mathematical discussions. If you watch this repo, you should be notified by email if there are new posts. You can watch it, by clicking this button:

grafik