**MSC:**- 14F05 Vector bundles, sheaves, related construction, See Also {18F20, 32Lxx, 46M20}
- 14H60 Vector bundles on curves, See also {14F05}

generalized sheaves on a smooth algebraic curve, which contains Stuhler's category of

generalized vector bundles or equivalently Seshadri's category of vector bundles with

parabolic structures. Furthermore, a cohomology theory for these generalized sheaves

analogous to the Zariski cohomology of coherent sheaves is discussed and used to

compute Ext-groups of generalized vector bundles. As an example, it is shown that the

Grothendieck decomposition theorem for vector bundles on $\mathbb P^1$ does not hold

for vector bundles with parabolic structures concentrated at three points.