A Gabriel Theorem for Coherent Twisted Sheaves

Abstract

We give a generalization of Gabriel's Theorem on coherent sheaves to the case of coherent twisted sheaves on a smooth variety X over a field k. We show that the category Coh(X,\alpha) determines the scheme structure of X for \alpha in the Brauer group of X, and that any equivalence between Coh(X,\alpha) and Coh(Y,\beta) induces an isomorphism between X and Y. In conclusion we prove the saturatedness of D^b(X,\alpha).