# 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!

## 7 thoughts on “Affineness results”

1. Matsushima’s criterion / Richardson’s theorem: A categorical quotient of an action of a linear group scheme on an affine scheme is affine if the group scheme is geometrically reductive. Conversely, a closed subgroup scheme of a general linear group (scheme) is geometrically reductive if and only if the quotient (left coset) space is affine.

2. A finite type complex scheme is affine if and only if the associated complex analytic space is Stein.

3. Every quasicompact, quasiseparated scheme \$X\$ of dimension 0 is affine. Proof:
(1) If \$U\$ is affine zero-dimensional, then \$\vert U\vert\$ is compact Hausdorff, so each quasicompact open of \$U\$ is closed, in particular affine.
(2) In general, we may assume by induction that \$X=U\cup V\$, with \$U\$ and \$V\$ affine open. It follows fro (1) that \$U\cap V\$ is clopen in \$U\$ and affine, so \$X=(U\setminus V)\coprod V\$ is affine.

4. Complements to the above:
It follows that a quasiseparated scheme X of dimension 0 is ind-quasi-affine (in particular separated). In fact one checks that if X is reduced (i.e. absolutely flat), the canonical map to its affine hull X’ is an open immersion, and X’ is absolutely flat.
Also, all this extends to algebraic spaces, using Gruson-Raynaud (aka Tag 07ST).

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