Random thoughts

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!

2 thoughts on “Random thoughts”

1. Johan, can you provide links to where you give a proof of (II) and (IV)? I’m curious to see if the proofs I have in mind are the same as yours. I’m not sure why (III) occurs between (II) and (IV) since it doesn’t seem relevant to the implication (II) ==> (IV).

• Yeah, I should have put in links and I should have ordered it (I), (III), (V), (II), and (IV). Currently we have
(I) is Algebra, Lemma 32.3 (Tag 00I5),
(II) is More on Morphisms, Lemma 13.3 (Tag 054Y),
(III) is Algebra, Lemma 86.4 (Tag 0539),
(IV) is More on Morphisms, Lemma 14.5 (Tag 0559) for the irreducible case and More on Morphisms, Lemma 15.4 (Tag 055G) for the connected case, and finally
(V) is Morphisms, Lemma 6.4 (Tag 02NE).

Frequent users of the stacks project will be aware that the numbering can change without warning and that only the tags are guaranteed to keep pointing to the same results.