affine coordinate ring in a sentence

1. There is a unique affine structure on this maximal spectrum that is compatible with the filtration on the affine coordinate ring.
2. We may define the function field of " V " to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
3. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate ring of " U ", and that a rational function on all of " V " consists of such local data which agree on the intersections of open affines.
4. It's difficult to find affine coordinate ring in a sentence.