Let X be a quasi-compact and quasi-separated algebraic space such that for every quasi-coherent O_X-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 D_QCoh(O_X). Let a ≤ b. The following are equivalent

1. E has tor amplitude in [a,b], and
2. for all F in QCoh(O_X) we have Hⁱ(E ⊗^L F)=0 for i not in [a,b].

See Tag 08IL.

Let A —> B be a local homomorphism of Noetherian local rings. Assume A —> B is formally smooth in the m_B-adic topology. Then A —> B is flat. See Tag 07NP.

PS: Of course much more is true, see Tag 07NQ.

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,O_X) is the integral closure of k_i in Γ(X,O_X). See Tag 04MK.