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I am reading page 255 of Qing Liu and he claims that if $U,V$ are affine open subsets of a scheme $X$, then $\mathcal{O}_X(U) \to \mathcal{O}_X(V)$ is a flat morphism. Why is this necessarily the case? There are no finiteness assumptions or anything right now on $X$.

If $V = \operatorname{Spec} A_f$ and $U = \operatorname{Spec} A$ then the restriction map is just the canonical map $A \to A_f$ which is flat by exactness of localization. What about the general case of $V$ not necessarily being a basic open set?

2 Answers2

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While reading page 255, your question is answered on page 136. Namely, we have the usual definition 3.1 of flatness. Then Proposition 3.3 tells us that open immersions are flat (which is trivial) and that a morphism of affine schemes $X \to Y$ is flat iff $\mathcal{O}(Y) \to \mathcal{O}(X)$ is a flat homomorphism (which is already proven on page 11 in Corollary 2.15). Hence, if $X \to Y$ is an open immersion of affine schemes, then $\mathcal{O}(Y) \to \mathcal{O}(X)$ is flat.

Remark: If $X \to Y$ is an open immersion between arbitrary schemes, then $\mathcal{O}(Y) \to \mathcal{O}(X)$ doesn't have to be flat.

Martin Brandenburg
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  • Thanks Martin for your answer. Indeed I wasn't able to answer my question because I wasn't aware that $\phi: A \to B$ is flat iff for every prime $\mathfrak{p}$ of $B$, $A_{\phi^{-1}(\mathfrak{p}} \to B_{\mathfrak{p}}$ is flat. –  Oct 05 '13 at 08:58
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I think this is just commutative algebra (as are many things in scheme theory!). Basically, a homomorphism of commutative rings $\phi:A\to B$ is flat if and only if the induced homomorphisms $A_{\mathfrak{\phi^{-1}(p)}}\to B_{\mathfrak{p}}$ are flat for all prime ideals $\mathfrak{p}$ of $B$. (If you don't know the proof of this, then it's a good exercise to do!)

In our case, these induced homomorphisms are actually isomorphisms so must be flat.

I hope that helps!

Amitesh Datta
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  • Dear Amitesh, indeed this criterion is new to me! I do know that $\phi : A \to B$ is flat iff $B$ is flat as an $A$-module iff for *every prime ideal $\mathfrak{q}$ of $A$*, we have $B_{\mathfrak{q}} = B \otimes_A A_{\mathfrak{q}}$ is flat as an a $A_{\mathfrak{q}}$-module. –  Oct 05 '13 at 08:55
  • And the fact you state is true: It is Corollary 2.15 of Qing Liu. Funny enough I have never come across this criterion before in my life, and I don't remember seeing it in Atiyah-Macdonald. –  Oct 05 '13 at 08:59
  • Dear @user38268, that's true for sure but I don't see how that can help us here. After all, the induced homomorphisms on stalks are only defined at points of $V$ as you observe above. Matsumura's Commutative Algebra contains lots of interesting results about flatness (including, I believe, this one though it's been a while since I've read Matsumura). – Amitesh Datta Oct 05 '13 at 09:00
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    It's Ben here. I now have even more motivation to read Matsumura: Atiyah-Macdonald is simply not enough; I can't believe I wasn't even aware of these equivalent properties of flat-homomorphisms! –  Oct 05 '13 at 09:02
  • Dear Ben, do you know the equational criterion for flatness? That's another basic result in (if I remember correctly) the first page of Matsumura's text! (It's also in the Stacks Project: http://stacks.math.columbia.edu/tag/00HK.) Also, faithful flatness is an important concept not covered in Atiyah and Macdonald. – Amitesh Datta Oct 05 '13 at 09:07
  • @Amitesh: Exercises 16 and 17 in chapter 5. Also exercise 12 in chapter 7, exercises 7 and 12 in chapter 10. There seems to be a typo in the Index which says that faithfully flatness is introduced on page 29, where flatness is defined. But this is not the case. – Martin Brandenburg Oct 05 '13 at 11:14
  • @Martin Yes, of course! I completely forgot about this as it has been a while since I've read Atiyah and Macdonald. Thanks for pointing this out! – Amitesh Datta Oct 05 '13 at 13:11