Schematic HN stratification for families of principal bundles and lambda modules

Abstract

For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan reduction with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. This has the consequence that principal bundles of a given Harder Narasimhan type form an Artin stack. We also prove a similar result showing the existence of a schematic Harder-Narasimhan filtration for flat families of pure sheaves of $\Lambda$-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of ${\mathcal O}$-modules proved earlier by Nitsure. This again has the implication that $\Lambda$-modules of a fixed Harder-Narasimhan type form an Artin stack.