Let X be a quasi-compact and quasi-separated algebraic space such that for every quasi-coherent OX-module F we have H1(X, F) = 0. Then X is an affine scheme. See Tag 07V6.
There exist a zero dimensional local ring with a nonzero flat ideal. See Tag 05FZ.
Let X be a quasi-separated algebraic space. Let E be an object of DQCoh(OX). Let a ≤ b. The following are equivalent
- E has tor amplitude in [a,b], and
- for all F in QCoh(OX) we have Hi(E ⊗L F)=0 for i not in [a,b].
See Tag 08IL.
Let A be a ring. Let I ⊂ J ⊂ A be ideals. If M is J-adically complete and I is finitely generated, then M is I-adically complete. See Tag 090T.
Let X be a scheme. Let a : X —> Spec(k1) and b : X —> Spec(k2) be morphisms from X to spectra of fields. Assume a,b are locally of finite type, and X is reduced, and connected. Then we have k′1 = k′2, where k′i ⊂ Γ(X,OX) is the integral closure of ki in Γ(X,OX). See Tag 04MK.