Abstract
Let $R$ be a regular local ring. Let $\mathbf {G}$ be a reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that a principal ${\mathbf {G}}$-bundle over $R$ is trivial if it is trivial over the quotient field of $R$. The conjecture is known when $R$ contains a field. We prove the conjecture for a large class of regular local rings
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