A Gabriel Theorem for coherent twisted sheaves

The aim of this work is to give a generalization of Gabriel's Theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes. We start by showing that we can recover a noetherian scheme X from the category Coh(X,\alpha) of coherent \alpha-twisted sheaves over X, where \alpha lies in the cohomological Brauer group of X. This follows from the bijective correspondence between closed subsets of X and Serre subcategories of finite type of Coh(X,\alpha). Moreover, any equivalence between Coh(X,\alpha) and Coh(Y,\beta), where X and Y are noetherian schemes, and \alpha\in Br'(X), \beta\in Br'(Y) induces an isomorphism between X and Y. © 2008 Springer-Verlag.