# Affineness results

For whatever reason I really enjoy results that tell us certain schemes are affine. Here is a list of a number of results of this nature in the Stacks project (but only those which deal with schemes — there are analogues of most of these results when we look at algebraic spaces and algebraic stacks):

Tag 02O0 A scheme whose underlying space is finite discrete is affine.

Tag 01PV The nonvanishing locus of a section of a line bundle on an affine scheme is affine.

Tag 0C3A Let Y be a locally closed subscheme of an affine scheme X and assume there is an affine open U of X such that Y ∩ U is affine and such that Y ∖ U is closed in X. Then Y is affine.

Tag 04DE If X → Y is a homeomorphism onto a closed subset of the affine scheme Y then X is affine.

Tag 01XF Vanshing of higher cohomology for quasi-coherent modules implies affine.

Tag 0EBE If X is quasi-affine and H^i(X, O_X) = 0 for i > 0 then X is affine.

Tag 0EBR Suppose you have a reflexive rank 1 module L over a local ring A and a section s of L such that s^n is contained in mA L[n]. Then the locus where s doesn’t vanish is affine. This generalizes the case of invertible modules mentioned above.

Tag 05YU If X → Y is surjective and integral (for example finite) and X is affine, then Y is affine.

Tag 09NL If a scheme X is the union of finitely many affine closed subschemes, then X is affine.

Tag 0A28 If X is a curve and not proper, then X is affine.

Tag 0F3R If f : X → Y is a morphism of affine schemes which has a positive weighting w, then the set V of points y of Y such that the total weight over y is maximal is an affine open of Y. For example, if f is etale, then V is the maximal open of Y over which f is finite etale. Other cases where one has a weighting are discussed in Lemmas Tag 0F3D and Tag 0F3E

Tag 0EB7 The complement of a 1 dimensional closed subset of the spectrum of a 2 dimensional normal excellent Noetherian local ring is affine.

Tag 0ECD Let f : X → Y be a finite type morphism of excellent affine schemes over a field with X normal and Y regular. Then the locus V in X where f is etale is affine. (This should be true without assuming Y to be over a field.) This result is a strengthening of purity of ramification locus which itself is a result of Gabber you can find in section Tag 0EA1.

I hope you enjoy this kind of result as well! If you know addtional results of this nature, please leave a comment or send me an email. Thanks!