Topology and purity for torsors (Q273496)

Summary: We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for \(\mathrm{PGL}_p\)-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 \textit{J. L. Colliot-Thelene} and \textit{J. J. Sansuc} [Math. Ann. 244, 105--134 (1979; Zbl 0418.14016)], for all \(\mathrm{PGL}_p\). 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.