Random thoughts on material I added to the stacks project lately:

(I) Suppose you have a ring map φ: A —> B with the following properties: (1) Ker(φ) is locally nilpotent, and (2) for every x in B there exists a n > 0 such that x^n is in Im(φ). Then it is true that Y = Spec(B) —> Spec(A) = X is a homeomorphism, but it is not true in general that Y —> X is a universal homeomorphism. A counter example is where A is a non-algebraically closed field which is an algebraic extension of F_p and B is the algebraic closure of A.

(II) Let f : X —> Y be a morphism of finite type where Y is integral with generic point η. Suppose Z is a closed subscheme of X such that Z_\eta = X_\eta set theoretically. Then there exists a nonempty open V ⊂ Y such that Z_V = X_V set theoretically. (In the Noetherian case this is pretty straightforward.)

(III) A torsion free module over a valuation ring is flat. (If you don’t know how to prove this then it is a nice exercise for when you’re in the shower.)

(IV) Let f : X —> Y is a morphism of finite type where Y is integral with generic point η. If X_η is geometrically irreducible, then there exist a nonempty open V ⊂ Y such that all fibres X_y, y ∈ V are geometrically irreducible. Same with geometrically connected.

(V) Let f : X —> Y be a quasi-compact morphism of schemes. Suppose η ∈ Y is a generic point of an irreducible component of Y which is not in the image of f. Then there exists an open neighborhood V ⊂ Y of η such that f^{-1}(V) is empty.

Let me know if any of these assertions are wrong… thanks!