Let A —> B be a ring map such that B ⊗_A B —> B is flat. Let N be a B-module. If N is flat as an A-module, then N is flat as a B-module. See Tag Tag 092C.

Let R be a ring. Let x ∈ R. Assume

1. R is a normal Noetherian domain,
2. R/xR is a Japanese domain,
3. R = lim R/xⁿR is complete with respect to x.

Then R is Japanese. See Tag 032P.

Let A be a valuation ring. Let A→B be a ring map of finite type. Let M be a finite B-module.

1. If B is flat over A, then B is a finitely presented A-algebra.
2. If M is flat as an A-module, then M is finitely presented as a B-module.

See Tag 053E.

PS: Much more is true, see the this chapter in the stacks project. The proof of the lemma above however is quite easy.

Let A be a Grothendieck abelian category. Then

1. D(A) has both direct sums and products,
2. direct sums are obtained by taking termwise direct sums of any complexes,
3. products are obtained by taking termwise products of K-injective complexes.

See Tag 07D9.

Let A —> B be a ring map. Assume

1. A ⊂ B is an extension of domains,
2. A is Noetherian,
3. A —> B is of finite type, and
4. the extension f.f.(A) ⊂ f.f.(B) is finite.

Let p ⊂ A be a prime such that dim(A_p) = 1. Then there are at most finitely many primes of B lying over p. See Tag 02MA.

Let R —> S be a ring map. Let p ⊂ R be a prime. Assume that

1. there exists a unique prime q ⊂ S lying over p, and
2. either
   1. going up holds for R —> S, or
   2. going down holds for R —> S and there is at most one prime of S above every prime of R.

Then S_p=S_q. See Tag 00EA.

Let X be a quasi-compact and separated algebraic space. Let U be an affine scheme, and let f : U —> X be a surjective étale morphism. Let d be an upper bound for the size of the fibres of |U| —> |X|. Then for any quasi-coherent O_X-module F we have H^q(X,F)=0 for q ≥ d. See Tag 072B.

Note: This is interesting even when X is a scheme.

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.