Unit-ideal certificate from a principal affine cover
= Unit-ideal certificate from a principal affine cover
{title2=$\sum_i g_if_i=1$}
If finitely many $X_{f_i}$ cover $X$, the morphism $\mathcal O_X^r\to\mathcal O_X$ with coefficients $f_i$ is onto. If its <kernel> has vanishing first <sheaf cohomology>, the <long exact sequence in sheaf cohomology> lifts the <global section> $1$ to coefficients $g_i$. This turns a geometric cover into a <unit ideal> in the ring of <global sections>.