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It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. https://rigtriv.wordpress.com/2008/04/09/locally-free-sheaves-and-vector-bundles/ for the algebraic variety case; a reference on this result would be nice also). I wonder if there is a similar correspondence between locally free sheaves of algebras and algebra bundles.

More precisely, I want to know, given a scheme $(X,\mathcal{O}_X)$ and a sheaf $\mathcal{A}$ of $\mathcal{O}_X$-algebras that is locally free and of constant and finite rank, whether or not there exists a bundle $A\to X$of algebras over $X$ such that $\Gamma_X(A)\cong\mathcal{A}$ as sheaves. I do not assume the sheaf $\mathcal{A}$ to be of commutative algebras, though I do asume associativity.

The proof in the above link works fine since I'm also assuming each algebra $\mathcal{A}_x$ to be finite-dimensional. The question I cannot answer is whether or not it is possible to have the gluing morphisms to be algebra isomorphisms.

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    $\begingroup$ As long as one is careful to phrase the correspondence between locally free sheaves and vector bundles as an equivalence of $\otimes$-categories, then the statement for algebras follows automatically since it is just the resulting correspondence between algebra objects in both categories. $\endgroup$ May 14, 2015 at 21:30

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