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Sixteenth post: More flatness

Posted by ravivakil under

Actual notes
[6] Comments
**The sixteenth post is the June 27 version in the usual place.** This post covers until 25.7 (although 25.8, still a work in progress, is also included). The next post should appear in late July.

**Status report.**

Most of the progress since the 15th post has been in tweaking, and gradually digesting people’s comments. I have a lot of valuable suggestions still to digest, and am steadily making progress.

We’re nearing the end of the year-long project. I foresee two more regular posts (one in late July one in mid to late August), by which time only a couple of chapters will remain unrevealed. Then I expect a period of disruption, as if all goes well, I’ll become a father again near the start of September (a second son expected). I’ll then be teaching from these notes this coming academic year, and really testing what can be done in a single year. I will also fill out the last couple of chapters, and continue to work through all the wise advice I’ve been given over the year on this site.

**For learners.**

The fifteenth post was longer than it should have been from the point of view of learning, so please go back and get comfortable with that material.

Here are some flatness exercises. (Section 25.5 is there thanks to Georges.)

25.3.A, 25.3.E, 25.4.A-C, 25.4.H, 25.5.A, 25.5.D, 25.5.E, 25.5.F, 25.6.A, 25.7.A, 25.7.C.

If you would like to work out an explicit example in detail, you may want to try the explicit flat limit at the end of 25.4. If you get stuck at any point, please let me know, and I will try to get you unstuck. (And as always, if you get stuck anywhere in these notes where you think a few words might help you out, please let me know — those few words should probably be added to the text!)

**For experts.**

I’m not sure what the right definition of “flat of relative dimension n” is. Currently I only discuss it in the locally Noetherian setting, because I want it to be closed under composition (with the obvious additivity of relative dimension), and my method of showing it uses Noetherian techniques. Can someone tell me (or point me to) the right definition (e.g. the location in EGA)?

June 28, 2011 at 12:02 pm

Dear Ravi,

maybe you could insert the caveat, in the form of a little exercise for example, that an open morphism is not necessarily flat. Since you have already mentioned the example Spec k[t] /(t) ->Spec k[t] /(t^2), this wouldn’t take up much space.

June 29, 2011 at 8:44 am

Good idea — now done!

July 13, 2011 at 4:48 pm

Hi,

I had a couple questions about section 18.6 and what conditions you needed for the theorems.

Firstly, in Theorem 18.4.2, you begin by assuming the curve C is affine and then taking an closed immersion into A^n. Do we need some kind of finite type condition to be able to do this? Is being locally Noetherian and a k-algebra good enough?

In the proof of Theorem 18.4.3, you state that the map between normalizations is a map between nonsingular irreducible curves. I don’t see why the normalization in this case has to be locally Noetherian (one of the hypothesis of being nonsingular).

Thanks for your help.

Lalit

July 15, 2011 at 9:40 am

Whoops! I forgot the finite type assumption. I’ve now added them (in the version to be posted, likely in about a week), to Theorem 18.4.1, Theorem 18.4.2, and Theorem 18.4.3(iv). Thanks for catching that!

July 21, 2011 at 9:02 am

Addendum to my question for experts: My meager understanding of “flatness of relative dimension n” seems to require even more: I want this notion to be preserved by base change. The easiest way I can think of is to add in finite presentation assumptions (implying finite type, as I’ve already forced myself to take Noetherian assumptions), and then use the transcendence interpretation of dimension. (To be honest, I still haven’t checked that if X is a pure-dimensional finite type k-scheme, then the same is true upon any extension of base field.

[Update March 25 2012 — now done.]) Any hints, thoughts, or references?May 4, 2012 at 11:30 am

Update: I now believe that the local finite presentation assumption is the right one to make, see

here. Everything now works, and this can’t be reasonably relaxed.