Does every open affine subscheme of an affine scheme have form $A_f$

Question

Let Spec${A}$ be an affine scheme, can every open affine subscheme be written as Spec$A_f$ for some $f$ in $A$?

score 3 · Accepted Answer · edited Apr 13 '17 at 12:58

3

No. Counterexamples are discussed at MO/7153. The example by Hailong Dao is really elementary: $A=k[X,Y,U,V]/(XY+UX^2+VY^2)$, then $D(X) \cup D(Y)$ is affine, but not basic-open.

edited Apr 13 '17 at 12:58

Community

1

answered Oct 23 '13 at 07:44

Martin Brandenburg

163,620

1

Why is $D(X) \cup D(Y)$ not basic-open? This was not answered over at MO. – red_trumpet Sep 17 '18 at 12:49

Does every open affine subscheme of an affine scheme have form $A_f$

1 Answers1