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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:
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.
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
synthetic-zariski/projective/tmp
copy all tex-files there and run
../../util/zar-rebase.sh ../../util/
latexmk -pdf -pvc main.tex
to produce the main.bbl
and check if the draft builds. .tar.gz
, so everything can be uploaded in one step, e.g.
tar -czv -f DRAFT.tar.gz *.tex *.cls *.sty main.bbl
... 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: