Well-Ordered Flag Spaces as Functors of Points

TL;DR

This paper extends the classical finite-dimensional flag space construction to an infinite-dimensional setting using functors of points, establishing group actions, Bruhat decomposition, and order relations in this new context.

Contribution

It introduces a new infinite-dimensional flag space as a functor of points, generalizing classical flag schemes and proving analogous geometric and combinatorial properties.

Findings

Defined an infinite-dimensional flag space as a functor of points.

Proved the existence of group actions and Bruhat decomposition.

Established the Bruhat order and closure relations in the infinite setting.

Abstract

Using Grothendieck's "functor of points" approach to algebraic geometry, we define a new infinite-dimensional algebro-geometric flag space as a $k$-functor (for $k$ a ring) which maps a $k$-algebra $R$ to the set of certain well-ordered chains of submodules of an infinite rank free $R$-module. This generalizes the well known construction of a $k$-functor that is represented by the classical (i.e. finite-dimensional) full flag scheme. We prove that as in the finite-dimensional case, there is an action of a general linear group on our flag space, that the stabilizer of the standard flag is the subgroup $B$ of upper triangular matrices, and that the Bruhat decomposition holds, meaning that our space is covered by the disjoint Schubert cells $\text{sh}(B \sigma B) / B$ indexed by permutations $\sigma$ of an infinite set. Finally, in the case of flags indexed by the ordinal $\omega + 1$, we define an analog of the Bruhat order on this infinite permutation group and prove that when $k$ is a domain, Ehresmann's closure relations still hold, i.e. that the closure $\overline{\text{sh}(B \sigma B) / B}$ is covered by the Schubert cells indexed by permutations smaller than $\sigma$ in the infinite Bruhat order.