Synthetic Algebraic Geometry in the Zariski-Topos

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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/articles:

• Foundations (latest pdf, arxiv, talk)
• Automorphisms of and line bundles on projective space (latest pdf, arxiv, talk)
• Synthetic stone duality (latest pdf, arxiv, old extended version)
• Differential Geometry/étale maps (pdf, arxiv, full notes pdf)

And the following drafts and notes:

• Čech-Cohomology (early draft pdf)
• Proper Schemes (early draft pdf)
• Topology of Synthetic Schemes (early draft pdf)
• $\mathbb A^1$-homotopy theory (early draft pdf)
• Algebraic spaces and stacks (very early draft pdf)
• More general topologies, in particular fppf (very early draft pdf)
• Calculations with (elliptic) curves and divisors (very early draft pdf)
• Preliminaries for Serre-Duality (very early draft pdf)
• Synthetic stone duality (pdf, summary)
• Cohomology and homotopy theory in synthetic stone duality (very early draft pdf)
• Notes on a model for synthetic stone duality (very early draft pdf)
• Finite schemes in SAG (very early draft pdf)
• Random Facts, i.e. a collection of everything that still needs to find a good place (very early draft pdf)
• Collection of exercises (pdf with exercise-ideas)

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
  

• Login to the arxiv
• Fill in forms - usually we choose the arxiv perpetual license.
• upload the tar
• Fill in more forms - for the foundations we used the following MSC subject classification:

  MSC-class: 14A99 (Primary), 03B38, 18N99 (Secondary)
  

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