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Grassmann sheaves and the classification of vector sheaves

Papatriantafillou M. H., Vassiliou E.

Demonstratio Mathematica

2013

Vol. 46, nr 2

263--270

We classify vector sheaves (an abstraction of vector bundles) by means of a universal Grassmann sheaf. This is done in three steps. Given a sheaf of unital commutative and associative algebras A, we first 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 (a locally free A-module) over a paracompact space is a subsheaf of A∞. In the last step, the foregoing considerations lead to the construction of a universal Grassmann sheaf GA(n), whose global sections classify vector sheaves of rank n over a paracompact space. Note that a homotopy classification is not applicable in this context.

On the cohomology and geometry of principal sheeaves

Vassiliou E.

Demonstratio Mathematica

2003

Vol. 36, nr 2

289--306

We study the cohomological classification of principal sheaves, the latter being defined in a slightly different way than in [6], a fact allowing to consider on them geometrical objects like connections. The classification of vector sheaves (studied in [10]) is now a corollary of the classification of their principal sheaves of frames. In particular, principal sheaves with an abelian structural sheaf, equipped (the former) with a connection, admit a hypercohomological classification generalizing that of Maxwell fields given in [10].