Topology and Purity for Torsors

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $\PGL_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all $\PGL_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.

2010 Mathematics Subject Classification:

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