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Favorite properties of k-schemes that can be checked after base change to the algebraic closure

Posted by ravivakil under

Big lists
[18] Comments
What are your favorite properties of varieties over a field *k* (or more generally finite type *k*-schemes) that can be checked after base change to the algebraic closure of *k*? This is of course secretly about faithfully flat descent, but my question is partially pedagogical.

Sometimes the implication goes only one way (e.g. if something is true over the algebraic closure, then it is true for k); please be clear on that.

I would like later to add references, so people can know not just what is true, but also why it is true. If you know references off the top of your head (even somewhat vague ones, such as “I remember a nice proof in [reference here]”, please post them).

For simplicity, please put one per comment, so people can respond and comment further.

June 13, 2011 at 9:50 am

smoothness of a morphism

June 13, 2011 at 9:50 am

surjectivity of a morphism

June 13, 2011 at 9:51 am

one way: injectivity of a morphism (and this implies universal injectivity). Example that it doesn’t go both ways: , a map of finite type schemes over the reals.

June 13, 2011 at 9:51 am

nonsingularity (only one way; both ways if k is perfect)

June 13, 2011 at 9:51 am

of a quasicoherent sheaf

June 13, 2011 at 9:51 am

pure dimension n

June 13, 2011 at 9:52 am

normality

June 13, 2011 at 9:53 am

factoriality? (I would expect this to be a one-way implication, and to be an example of a UFD that is no longer a UFD upon base change to .)

June 15, 2011 at 2:41 pm

is not a UFD, but its base change to is.

June 15, 2011 at 2:42 pm

Is there a quick reference/proof?

June 16, 2011 at 6:34 am

Let . is a UFD since it’s isomorphic to by . On the other hand, in R, the height one prime is not principal. Suppose . Let X be the closure of . Since f is a real polynomial, if it vanishes on a point of X, it must also vanish on the complex conjugate point, so it must vanish at an even number of points of X which are not fixed by complex conjugation. Since X has degree 2, f must also vanish at an even number of points overall, so it cannot have a simple zero at $(0,1)$ but not vanish at any other point of .

I think that at least some of this is also in Mumford’s Red Book.

Of course, R is still locally factorial, since it’s one-dimensional and nonsingular.

June 21, 2011 at 7:48 am

This is a nice example, and gives a interesting contrast (which I’ll now explain). By factoriality, I meant what you mean by “local factoriality”, not UFD-ness (“global factoriality?”), and I believe that the stalk at the origin of *is* a UFD, and *know* that the stalk of is *not* a UFD, because there is a height 1 ideal that is not principal — this height 1 prime ideal “is not present” over (its double *is* principal).

And in your examples, we have something very similar over $\mathbb{R}$ that *is not* a UFD (because there is a height 1 ideal that is not principal), that becomes principal over $\mathbb{C}$, and hence ring *becomes* a UFD — the opposite behavior!

June 13, 2011 at 9:54 am

reducedness (one way); geometric reducedness (both ways)

June 13, 2011 at 9:54 am

irreducibility (one way); geometric irreducibility (both ways)

June 19, 2011 at 2:50 pm

Connected (one way); geometrically connected (both ways)

June 19, 2011 at 3:18 pm

Proper (both ways)

June 19, 2011 at 3:18 pm

Affine (both ways)

June 19, 2011 at 3:20 pm

Quasi-compact (both ways)