The Stacks project

Lemma 56.10.4. 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 20.40. 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 20.48.4 (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. Simlarly 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.


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