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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 parts:
- Foundations (latest pdf, arxiv)
- Čech-Cohomology (early draft pdf)
- Differential Geometry/étale maps (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)
- Automorphisms of and line bundles on projective space (very early draft pdf)
- More general topologies, in particular fppf (very early draft pdf)
- Calculations with (elliptic) curves and divisors (very early draft pdf)
- 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.
- 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$R[X] \to A$ ? - 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.) - 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?
-
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.
There are some recordings of talks from the last workshop on synthetic algebraic geometry. And there is a hottest talk on the foundations article.
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.
To put one of the drafts on the arxiv, we have to
- copy everything into one (temporary) folder: all tex-files, zariski.cls, zariski.sty from util and main.bbl.
- change the paths in zariski.cls and main.tex
- possibly change formulation from "This is a draft [...]"
- test by running latexmk
- put all the files into a
.tar.gz, so everything can be uploaded in one step
... is a good idea since we started to use the issue-tracker
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:
