# Κ-presentation

A morphism of finite presentation X —> S is a morphism which is (a) locally of finite presentation, (b) quasi-separated, and (c) quasi-compact.

Let κ be an infinite cardinal. What should be a morphism of κ-presentation? By analogy with the above I think it should be a morphism f : X —> S such that

1. for any affine opens U, V of X, S with f(U) ⊂ V the algebra O(U) is of the form O(V)[x_i; i ∈ I]/(f_j; j ∈ J) with |I|, |J| ≤ κ,
2. for any U, U’ affine open in X over an affine V of S the intersection U ∩ U’ can be covered by κ affine opens, and
3. for any affine V in S the inverse image f^{-1}(V) can be covered by κ affine opens.

It is my guess that all the usual things we prove for morphisms of finite presentation also hold for morphisms of κ-presentation. Namely, it should be enough to check the conditions over the members of an affine open covering of Y, the base change of a morphism of κ-presentation is a morphism of κ-presentation, etc. In particular, if should also be true that if {S_i —> S} is an fpqc covering and X_i —> S_i is the base change of f : X —> S, then

X —> S is of κ-presentation ⇔ each X_i —> S_i is of κ-presentation

Of course this is completely orthogonal to most of algebraic geometry and I hope you’ve already stopped reading several lines above (maybe when I used the key word “cardinal”). For those of you still reading let me indicate what prompted me to write this post. Namely, suppose that X, Y are schemes over a base S which are fpqc locally isomorphic. Then the above says that X and Y have roughly the same “size” (this is defined precisely in the chapter on sets in the stacks project).

As an application this tells us for example that given a group scheme G over S there is a set worth of isomorphism classes of principal homogeneous G-spaces over S! A principal homogeneous G-space is defined in the stacks project, as in SGA3, to be a pseudo G-torsor which is fpqc locally trivial — and note that the collection of fpqc coverings of S forms a proper class, which does not contain a cofinal subset!

Another potential application, internal to the stacks project and with notation and assumptions as in the stacks project, is that, given a group algebraic space G over S, it guarantees that the stack of principal homogeneous G-spaces form a stack in groupoids over (Sch/S)_{fppf}. Instead of working this out in detail in the stacks project I will for now put in a link to this blog post.

## 3 thoughts on “Κ-presentation”

1. BCnrd on said:

Johan, I was astonished by your conclusion about G-torsors for the fpqc topology. Is it really the case that your proposed definition of $\kappa$-presentation is a workable one, such as being able to check it using a much more restricted class of affine opens? This is relevant if you want to prove that this property is preserved under composition. Also, surely one has to prove (as must be done for finite presentation) that if $B$ is $\kappa$-presented over $A$ in the sense of being a quotient of a polynomial ring over $A$ in at most $\kappa$ variables modulo an ideal with at most $\kappa$ generators then necessarily *every* $A$-algebra map onto $B$ by a polynomial ring over $A$ in at most $\kappa$ variables has kernel with at most $\kappa$ relations. Is that obvious?

One reason it may not be obvious is that the argument for finite presentation uses the trick of passage to the noetherian case right here, so already you’d need a “new” proof for finite presentation if it is to work more generally. Anyway, so I think some care is needed before one should have confidence that this is a good notion without weird surprises.

• Johan on said:

Yes, I did briefly think this through and I think all of this does work with the _same_ arguments as in the finite presentation case. For the precise question you ask: You have B = A[x_k; k \in K]/I and also B = A[y_l; l \in L]/J where the index sets K and L have cardinality kappa, and where I can be generated by kappa elements. Then you consider the presentation B = A[x_k, y_l]/W and you argue that W can be generated by generators of I and elements of the form y_l – f_l with f_l in A[x_k; k \in K]. Then the image of W in A[y_l; l \in L] is J and you win. Etc, etc.

Of course the devil is in the details, but it is my experience that these arguments having to do with infinite “size” are typically easier than the corresponding arguments on finite type and finite presentation. I have already added a bunch of these. The most interesting one so far (for me) is the one about monomorphisms (which I found in the literature).

• Johan on said:

By the way, a nice example of a “big” group scheme where you can prove explicitly that the collection of isomorphism classes of torsors forms a set, is in the chapter on Examples, the section entitled “A torsor which is not a fppf torsor”.