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):<\/p>\n
Tag 02O0<\/a> A scheme whose underlying space is finite discrete is affine.<\/p>\n Tag 01PV<\/a> The nonvanishing locus of a section of a line bundle on an affine scheme is affine.<\/p>\n Tag 0C3A<\/a> 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 \u2229 U is affine and such that Y \u2216 U is closed in X. Then Y is affine.<\/p>\n Tag 04DE<\/a> If X → Y is a homeomorphism onto a closed subset of the affine scheme Y then X is affine.<\/p>\n Tag 01XF<\/a> Vanshing of higher cohomology for quasi-coherent modules implies affine.<\/p>\n Tag 0EBE<\/a> If X is quasi-affine and H^i(X, O_X) = 0 for i > 0 then X is affine.<\/p>\n Tag 0EBR<\/a> 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<\/sub> L[n]<\/sup>. Then the locus where s doesn’t vanish is affine. This generalizes the case of invertible modules mentioned above.<\/p>\n Tag 05YU<\/a> If X → Y is surjective and integral (for example finite) and X is affine, then Y is affine.<\/p>\n Tag 09NL<\/a> If a scheme X is the union of finitely many affine closed subschemes, then X is affine.<\/p>\n Tag 0A28<\/a> If X is a curve and not proper, then X is affine.<\/p>\n Tag 0F3R<\/a> 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<\/a> and Tag 0F3E<\/a><\/p>\n Tag 0EB7<\/a> The complement of a 1 dimensional closed subset of the spectrum of a 2 dimensional normal excellent Noetherian local ring is affine.<\/p>\n Tag 0ECD<\/a> 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<\/a>.<\/p>\n 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!<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading