Question

Hi there, I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with infinite dimensional general linear groups ? Is there a well-known, standard way to define its group scheme, just as in the finite dimensional case ? Or is this usually done using Ind-scheme constructions ? (And if so, can somebody describe it in a nutshell for, e.g., uncountable dimensions ?)

Thanks !!

Answer 1

You can write an infinite dimensional general linear group as a filtered colimit of finite dimensional general linear groups, where the filtering is over finite subsets of an infinite basis, ordered by inclusion. This process produces an ind-scheme, since each inclusion is a closed embedding. The basis can be uncountable.