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License: MIT License
Latex documentation of our understanding of the synthetic /internal theory of the Zariski-Topos
License: MIT License
Peter Arndt suggested the following classic trick:
There is a map to
The conditions are closed so this type of matrices is affine and all fibers of the map to
The latter implies this map is an
This extends to closed subsets of
For an open subset
Let
Formulated more in line with the A1-homotopy draft:
Is
Using reasoning from 2.2 in the A1-homotopy draft, it is enough to show that
We should write down what we know about those automorphism groups.
I think we should consider having a table somewhere listing all important definitions and theorems, together with a latex identifier (or issue) and a link to a formalization, if it exists.
One advantage should be, that definitions and usages are easy to find then, using the latex-identifier. On the other hand, those have to be longer then.
Another thing is, that if we start to point to formalization (which I think should always be done by pointing to a part of a file in a specific commit), we might be able to build something which is update-able in a not-so-painful way.
A subset
Then the inclusion
Until recently I thought a scheme of finite type is locally of the form
As far as I know, we do not have a common name of the structure sheaf of a scheme
There is one way to ask our base ring R to be of characteristic 0: Define the map
But this is a bit suspicious in light of this excerpt from @iblech 's thesis:
this might be problematic. So one concrete question is: Can we show that
Shall we make this repo BibTex-aware?
So, in algebraic geometry we're often interested in varieties which are usually defined as an integral, separated scheme of finite type over an (algebraically closed) field
Now, we know very well that
At the last hour of SAG-4, we had a discussion on names. The following was discussed in more detail:
We call a module bundle on a type weakly quasi-coherent (wqc), if its sections on opens are given by (algebraic) localization (in some sense). At some occasions we concluded that the name is not so great, since these module bundles are not really an analogue of quasi-coherent sheaves. I don't remember the details of these past discussions, but I would argue that in general, our theory (in particular wqc modules and their cohomology, pullbacks and push-forwards) diverts from the classic story, so I think it would be good to also divert with our names a bit more.
I think just calling the bundles "local module bundles" would be a good fit. Happy to hear opinions on that.
In Proposition 5.1.3 in the Foundations, it says
basic opens of the affine parts of
$X$
where
This is a sort of continuation to #18.
Does the Nisnevich
The nLab page on motivic homotopy theory suggests that if one is working in the right topos (i.e., Nisnevich), then localizing at
I wonder what one can make out of this.
Hugo suggested to me, it might be good to call an affine scheme
The proposition
We want to refer to axioms when proving theorems, ideally this should be a macro, e.g. '\Loc' and there should be a link.
Write down a proof that the map from the universe to its
Use that: For types A,B a line in
(joint idea with @MatthiasHu )
A type
What we call
The Segre-Embedding shows that (quasi-) projective schemes are closed under products. Is that also true for dependent sums? I don't know the classical answer but there might be a more or less immediate generalization of the classical argument using Zariski-choice and the Segre-Embedding.
In other words: Do we have
Using reasoning from 2.2 in the A1-homotopy draft, this would also imply that
Can we give a pointwise definition of flat maps between (affine) schemes?
This (hopefully) boils down to the following external question:
Let
Assuming a flat-modality, it should work to use David Jaz Myers idea in the good fibrations article, theorem 5.9 taking for X the type of torsors of crisply discrete groups, to show #17.
I remember that @fabianmasato and @mnieper discussed a proof with me, that maps from a connected projective scheme, i.e. a connected closed subset of some
Classically, we know that a flat morphism between varieties has a constant fibre dimension. Can we use this somehow to be able to define the notion of a flat scheme of dimension
At various points, we would like to quantify internally not over all types, but over "locally locally constant sheaves" (?). For example, naively we would say a general (not necessarily quasi-compact) open proposition is one of the form
But how about the following more general notion. (I think we discussed something like this at SAG 2 with @iblech.) We define what it means for a (0-truncated) type
I don't know if this is exactly the right definition. But my question is: can we use some definition like this to go beyond finiteness restrictions in SAG, to define general opens and perhaps general schemes (not necessarily finitely presented)?
It would be good to have a synthetic version of the Proj construction, as it would be useful for constructing projective schemes.
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