Replace "finite type" with "of finite presentation"

[Felix Cherubini]

Until recently I thought a scheme of finite type is locally of the form $\mathrm{Spec}(R[X_1,\dots,X_n]/(f_1,\dots,f_m))$ - but that is actually called scheme of finite presentation. So this should be wrong in quite a lot of places. We should change all drafts accordingly:

• A1-homotopy
• cech
• diffgeo
• elliptic
• foundations
• proper
• random-facts
• sheaves
• stacks
• topology
• slides

[Ingo Blechschmidt]

I agree, only in the Noetherian case do these coincide (and our generic ring R is never Noetherian, as else all rings would be). Good catch!

[Felix Cherubini]

Yes, good catch - not mine though ;-)

[Felix Cherubini]

I think I replaced that now everywhere.
I chose to also replace it in archived slides, since they might be reused.

[Xuanrui Qi]

I believe schemes of finite type are useful too. A scheme is finite type if it's locally $\mathbf{Spec} \ A$ where $R \rightarrow A$ is of finite type.

Stacks gives the definition: $R \rightarrow A$ is finite type if there is a surjection $R[x_1, ..., x_n] \rightarrow A$. This should be constructively valid, and I think is a workable definition.

[Marc Nieper-Wißkirchen]

I believe schemes of finite type are useful too. A scheme is finite type if it's locally Spec A where R→A is of finite type.

Stacks gives the definition: R→A is finite type if there is a surjection R[x1,...,xn]→A. This should be constructively valid, and I think is a workable definition.

In classical algebraic geometry, a scheme of finite type is locally a closed subset of affine space. This would not necessarily be true in our setting because a closed subset needs to be given by finitely many equations. Thus, I wonder whether the verbatim copy of the classical notion of finite type is really that much of interest.

[Felix Cherubini]

This issue was just about a mistake of mine in saying which external schemes we expect to match our internal notion of scheme. I have nothing against schemes of finite type in general ;-)

[Felix Cherubini]

But I agree with Marc, that it is unclear if we can say anything interesting about schemes of finite type in our current framework.

[Marc Nieper-Wißkirchen]

We might have to introduce infinitary logic to be able to talk sensibly about non-finite intersections of closed subsets. But that's just a very wild guess.

[Xuanrui Qi]

I don't think we need infinitary logic if we're not explicitly writing out the conjunctions. That said, it's probably impossible to construct explicitly any non-finitely-generated algebra and prove it to be so.

[Xuanrui Qi]

or in other words we could perhaps safely ignore this issue: it's very likely that any scheme we could reasonably define synthetically is of finite type externally.

[Ingo Blechschmidt]

That said, it's probably impossible to construct explicitly any non-finitely-generated algebra and prove it to be so.

Hm, what do you mean with that? For instance, the algebra $R[X_1,X_2,\ldots]$ and the power series algebra $R[[X]]$ (definable without infinitary language as the ring of functions $\mathbb{N} \to R[[X]]$) are provably not finitely generated.