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A generalization of central simple algebras over arbitrary commutative rings, or over a scheme
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An example of an Azumaya algebra that isn't free over its centre
It suffices to find a vector bundle $E$ on an affine variety $V$ such that the vector bundle $\mathcal{E}nd(E)$ is nontrivial: then this vector bundle gives a non-free, projective Azumaya algebra over …