  We develop a new form of patching that is both far-reaching and more
elementary than the previous versions that have been used in inverse Galois
theory for function fields of curves. A key point of our approach is to work
with fields and vector spaces, rather than rings and modules. After presenting
a self-contained development of this form of patching, we obtain applications
to other structures such as Brauer groups and differential modules.

Algebraic Geometry;Rings and Algebras