The Stacks project

Lemma 37.69.5. Consider a commutative diagram

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

of morphisms of schemes. Let $s \in S$ be a point. Assume

  1. $X \to S$ is locally of finite presentation and flat at points of $X_ s$,

  2. $f$ is proper,

  3. the fibres of $f_ s : X_ s \to Y_ s$ have dimension $\leq 1$ and $R^1f_{s, *}\mathcal{O}_{X_ s} = 0$,

  4. $\mathcal{O}_{Y_ s} \to f_{s, *}\mathcal{O}_{X_ s}$ is surjective.

Then there is an open $Y_ s \subset V \subset Y$ such that (a) $f^{-1}(V)$ is flat over $S$, (b) $\dim (X_ y) \leq 1$ for $y \in V$, (c) $R^1f_*\mathcal{O}_ X|_ V = 0$, (d) $\mathcal{O}_ V \to f_*\mathcal{O}_ X|_ V$ is surjective, and (b), (c), and (d) remain true after base change by any $Y' \to V$.

Proof. Let $y \in Y$ be a point over $s$. It suffices to find an open neighbourhood of $y$ with the desired properties. As a first step, we replace $Y$ by the open $V$ found in Lemma 37.69.2 so that $R^1f_*\mathcal{O}_ X$ is zero universally (the hypothesis of the lemma holds by Lemma 37.69.1). We also shrink $Y$ so that all fibres of $f$ have dimension $\leq 1$ (use Morphisms, Lemma 29.28.4 and properness of $f$). Thus we may assume we have (b) and (c) with $V = Y$ and after any base change $Y' \to Y$. Thus by Lemma 37.69.4 it now suffices to show (d) over $Y$. We may still shrink $Y$ further; for example, we may and do assume $Y$ and $S$ are affine.

By Theorem 37.15.1 there is an open subset $U \subset X$ where $X \to S$ is flat which contains $X_ s$ by hypothesis. Then $f(X \setminus U)$ is a closed subset not containing $y$. Thus after shrinking $Y$ we may assume $X$ is flat over $S$.

Say $S = \mathop{\mathrm{Spec}}(R)$. Choose a closed immersion $Y \to Y'$ where $Y'$ is the spectrum of a polynomial ring $R[x_ e; e \in E]$ on a set $E$. Denote $f' : X \to Y'$ the composition of $f$ with $Y \to Y'$. Then the hypotheses (1) – (4) as well as (b) and (c) hold for $f'$ and $s$. If we we show $\mathcal{O}_{Y'} \to f'_*\mathcal{O}_ X$ is surjective in an open neighbourhood of $y$, then the same is true for $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$. Thus we may assume $Y$ is the spectrum of $R[x_ e; e \in E]$.

At this point $X$ and $Y$ are flat over $S$. Then $Y_ s$ and $X$ are tor independent over $Y$. We urge the reader to find their own proof, but it also follows from Lemma 37.66.1 applied to the square with corners $X, Y, S, S$ and its base change by $s \to S$. Hence

\[ Rf_{s, *}\mathcal{O}_{X_ s} = L(Y_ s \to Y)^*Rf_*\mathcal{O}_ X \]

by Derived Categories of Schemes, Lemma 36.22.5. Because of the vanishing already established this implies $f_{s, *}\mathcal{O}_{X_ s} = (Y_ s \to Y)^*f_*\mathcal{O}_ X$. We conclude that $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is a map of quasi-coherent $\mathcal{O}_ Y$-modules whose pullback to $Y_ s$ is surjective. We claim $f_*\mathcal{O}_ X$ is a finite type $\mathcal{O}_ Y$-module. If true, then the cokernel $\mathcal{F}$ of $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is a finite type quasi-coherent $\mathcal{O}_ Y$-module such that $\mathcal{F}_ y \otimes \kappa (y) = 0$. By Nakayama's lemma (Algebra, Lemma 10.20.1) we have $\mathcal{F}_ y = 0$. Thus $\mathcal{F}$ is zero in an open neighbourhood of $y$ (Modules, Lemma 17.9.5) and the proof is complete.

Proof of the claim. For a finite subset $E' \subset E$ set $Y' = \mathop{\mathrm{Spec}}(R[x_ e; e \in E'])$. For large enough $E'$ the morphism $f' : X \to Y \to Y'$ is proper, see Limits, Lemma 32.13.4. We fix $E'$ and $Y'$ in the following. Write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as the colimit of its finite type $\mathbf{Z}$-subalgebras. Set $S_ i = \mathop{\mathrm{Spec}}(R_ i)$ and $Y'_ i = \mathop{\mathrm{Spec}}(R_ i[x_ e; e \in E'])$. For $i$ large enough we can find a diagram

\[ \xymatrix{ X \ar[d] \ar[r]_{f'} & Y' \ar[d] \ar[r] & S \ar[d] \\ X_ i \ar[r]^{f'_ i} & Y'_ i \ar[r] & S_ i } \]

with cartesian squares such that $X_ i$ is flat over $S_ i$ and $X_ i \to Y'_ i$ is proper. See Limits, Lemmas 32.10.1, 32.8.7, and 32.13.1. The same argument as above shows $Y'$ and $X_ i$ are tor independent over $Y'_ i$ and hence

\[ R\Gamma (X, \mathcal{O}_ X) = R\Gamma (X_ i, \mathcal{O}_{X_ i}) \otimes ^\mathbf {L}_{R_ i[x_ e; e \in E']} R[x_ e; e \in E'] \]

by the same reference as above. By Cohomology of Schemes, Lemma 30.19.2 the complex $R\Gamma (X_ i, \mathcal{O}_{X_ i})$ is pseudo-coherent in the derived category of the Noetherian ring $R_ i[x_ e; e \in E']$ (see More on Algebra, Lemma 15.64.17). Hence $R\Gamma (X, \mathcal{O}_ X)$ is pseudo-coherent in the derived category of $R[x_ e; e \in E']$, see More on Algebra, Lemma 15.64.12. Since the only nonvanishing cohomology module is $H^0(X, \mathcal{O}_ X)$ we conclude it is a finite $R[x_ e; e \in E']$-module, see More on Algebra, Lemma 15.64.4. This concludes the proof. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E7J. Beware of the difference between the letter 'O' and the digit '0'.