For a tensor triangulated Z/p-category K, with spectrum Spc(K), we construct an injective group homomorphism H~ 1(Spc(X), ) Z[1/p] Pic(K) Z[1/p], where Pic(K) is the group of -invertible objects of X. In modular representation theory, we prove that this homomorphism induces a rational isomorphism between the Picard group of the projective support variety and the group of endotrivial representations.
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