Schematic Harder–Narasimhan stratification for families of principal bundles and Λ-modules

Abstract.

Let G be a reductive algebraic group over a field k of characteristic zero, let X → S be a smooth projective family of curves over k, and let E be a principal G bundle on X. The main result of this note is that for each Harder–Narasimhan type τ there exists a locally closed subscheme S ^τ(E) of S which satisfies the following universal property. If f : T → S is any base-change, then f factors via S ^τ(E) if and only if the pullback family f ^∗ E admits a relative canonical reduction of Harder–Narasimhan type τ. As a consequence, all principal bundles of a fixed Harder–Narasimhan type form an Artin stack. We also show the existence of a schematic Harder–Narasimhan stratification for flat families of pure sheaves of Λ-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of 𝓞-modules proved earlier by Nitsure. This again has the implication that Λ-modules of a fixed Harder–Narasimhan type form an Artin stack.