Grassmann sheaves and the classification of vector sheaves

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Given a sheaf of unital commutative and associative algebras A, first we construct the k-th Grassmann sheaf Gₐ(k,n) of Aⁿ whose sections induce vector subsheaves of Aⁿ of rank k. Next we show that every vector sheaf over a paracompact space is a subsheaf of A∞. Finally, applying the preceding results to the universal Grassmann sheaf Gₐ(n), we prove that vector sheaves of rank n over a paracompact space are classified by the global sections of Gₐ(n).