Blown-up \v{C}ech cohomology and Cartan's Theorem B on real algebraic varieties

TL;DR

This paper develops a new form of cech cohomology for certain sheaves on real algebraic varieties, proving foundational theorems and applications like solving the Cousin problem after blow-ups.

Contribution

It introduces blown-up cech cohomology for specific sheaves on real affine varieties, establishing key properties and applications.

Findings

Established long exact cohomology sequence.

Proved Cartan's Theorem B in this context.

Provided universal solution to the first Cousin problem after blow-up.

Abstract

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).