  We introduce a concept of blown-up \v{C}ech cohomology for coherent sheaves
of homological dimension $\leq 1$ and some quasi-coherent sheaves on a
non-singular real affine variety. Its construction involves a directed set of
multi-blowups. We establish, in particular, long exact cohomology sequence and
Cartan's Theorem B. Finally, some applications are provided, including
universal solution to the first Cousin problem (after blowing up).

Algebraic Geometry