The Stacks project

Lemma 57.9.4. Let $X$ be a regular Noetherian scheme of dimension $d < \infty $. Then

  1. for $\mathcal{F}$, $\mathcal{G}$ coherent $\mathcal{O}_ X$-modules we have $\mathop{\mathrm{Ext}}\nolimits ^ n_ X(\mathcal{F}, \mathcal{G}) = 0$ for $n > d$, and

  2. for $K, L \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and $a \in \mathbf{Z}$ if $H^ i(K) = 0$ for $i < a + d$ and $H^ i(L) = 0$ for $i \geq a$ then $\mathop{\mathrm{Hom}}\nolimits _ X(K, L) = 0$.

Proof. To prove (1) we use the spectral sequence

\[ H^ p(X, \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{p + q}_ X(\mathcal{F}, \mathcal{G}) \]

of Cohomology, Section 20.43. Let $x \in X$. We have

\[ \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})_ x = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x) \]

see Cohomology, Lemma 20.51.4 (this also uses that $\mathcal{F}$ is pseudo-coherent by Derived Categories of Schemes, Lemma 36.10.3). Set $d_ x = \dim (\mathcal{O}_{X, x})$. Since $\mathcal{O}_{X, x}$ is regular the ring $\mathcal{O}_{X, x}$ has global dimension $d_ x$, see Algebra, Proposition 10.110.1. Thus $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$ is zero for $q > d_ x$. It follows that the modules $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})$ have support of dimension at most $d - q$. Hence we have $H^ p(X, \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^ q(\mathcal{F}, \mathcal{G})) = 0$ for $p > d - q$ by Cohomology, Proposition 20.20.7. This proves (1).

Proof of (2). We may use induction on the number of nonzero cohomology sheaves of $K$ and $L$. The case where these numbers are $0, 1$ follows from (1). If the number of nonzero cohomology sheaves of $K$ is $> 1$, then we let $i \in \mathbf{Z}$ be minimal such that $H^ i(K)$ is nonzero. We obtain a distinguished triangle

\[ H^ i(K)[-i] \to K \to \tau _{\geq i + 1}K \]

(Derived Categories, Remark 13.12.4) and we get the vanishing of $\mathop{\mathrm{Hom}}\nolimits (K, L)$ from the vanishing of $\mathop{\mathrm{Hom}}\nolimits (H^ i(K)[-i], L)$ and $\mathop{\mathrm{Hom}}\nolimits (\tau _{\geq i + 1}K, L)$ by Derived Categories, Lemma 13.4.2. Similarly if $L$ has more than one nonzero cohomology sheaf. $\square$

Comments (3)

Comment #6479 by Noah Olander on

It seems not completely trivial that 0A7U implies what you say it does, since 0A7U is really only about the case G = O_X.

Comment #6480 by on

Oops, yes indeed. Will fix it next time I go through all the comments.

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 0FZ3. Beware of the difference between the letter 'O' and the digit '0'.