Mixed-characteristic Grothendieck–Serre conjecture beyond the unramified cases

Establish the Grothendieck–Serre conjecture for reductive group schemes over regular local rings in the unresolved mixed-characteristic cases beyond the previously known unramified situations, including constant and totally isotropic groups.

Background

The paper studies the Grothendieck–Serre conjecture, which asserts that a generically trivial torsor under a reductive group scheme over a regular local ring is trivial. Before this work, the mixed-characteristic case was known only in several special unramified settings, notably for constant groups and totally isotropic groups.

The unresolved problem concerns extending the result to the remaining mixed-characteristic regular local rings and reductive group schemes. The paper proves the conjecture for all unramified regular local rings, thereby resolving that specific class while leaving the broader mixed-characteristic situation open.