Tag: covers

Many apologies if this is totally standard! I couldn’t find it in the literature.

Background definitions:

• A presheaf $X: textbf{Aff}^text{op} to textbf{Set}$ is an ind-scheme if it is a filtered colimit of schemes, where we regard a scheme as a presheaf via the Yoneda embedding (and then restrict to affine schemes), and the colimit is taken in the category $text{Psh}(textbf{Aff})$ of presheaves (i.e. the colimit is taken “pointwise”). One can impose the condition that all of the transition maps are closed immersions, and then call the result a strict ind-scheme.

• Let’s say that $X$ is an ind-affine ind-scheme if it is a filtered colimit of affine schemes. (Side question: Is this the right terminology? It seems there are several distinct notions with this name in the literature.)

Context:

• It is well known that a presheaf $X in text{Psh}(textbf{Aff})$ is represented by a scheme if and only if it satisfies the following two conditions: 1) $X$ is a Zariski sheaf and, 2) There is a covering of $X$ by open affine subfunctors.

• Every ind-scheme is a Zariski sheaf (in fact satisfies the sheaf condition for the fpqc topology).

• The affine Grassmannian (which is an ind-scheme) admits an open cover by ind-affine ind-schemes. See, e.g. the extremely helpful paper by Timo Richarz “Basics on Affine Grassmannians”.

Question:

Is there a similar characterization for (strict or otherwise) ind-schemes? I.e. is something like the following true: a presheaf $X$ is an ind-scheme if and only if 1) $X$ is a Zariski sheaf and, 2) $X$ admits a covering by open subfunctors which are ind-affine ind-schemes? Is it at least sufficient? If not, is there a name for the presheaves that satisfy this property?