On a theorem of Harder

TL;DR

This paper generalizes Harder’s theorem by proving that for a Dedekind domain $A$ and a simply connected semisimple group $G$ that is strictly isotropic over the fraction field $L$, the map $H^1_{et}(A,G) -> H^1_{et}(L,G)$ has trivial kernel, yielding $H^1_{Zar}(A,G)=1$ and extending the result to isotropic groups. The proof employs a parabolic lifting argument, Levi decompositions via a cocharacter $eta: G_m o G$, and a patching framework that combines local data with global group-theoretic decompositions, avoiding reliance on strong approximation results. The paper also connects these findings to Nisnevich’s conjecture, deriving two concrete cases where $H^1_{et}$-vanishing over function-field localizations implies $H^1_{Zar}$-triviality, thereby advancing the equicharacteristic and mixed-characteristic instances of the conjecture. Overall, the work provides a structural approach to principal bundles over Dedekind domains in the isotropic, simply connected semisimple setting and extends techniques for patching and cocharacter methods in this context.

Abstract

We prove that for any simply connected isotropic reductive group G over a Dedekind domain D, any Zariski-locally trivial principal G-bundle over D is trivial. The corresponding result for quasi-split groups was proved in 1967 by G. Harder.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['th:main']}
• Conjecture 3.1
• Conjecture 3.2
• Theorem 3.3
• proof