The Stacks project

48.34 Lichtenbaum's theorem

The theorem below was conjectured by Lichtenbaum and proved by Grothendieck (see [Hartshorne-local-cohomology]). There is a very nice proof of the theorem by Kleiman in [Kleiman-Lichtenbaum]. A generalization of the theorem to the case of cohomology with supports can be found in [Lyubeznik-Lichtenbaum]. The most interesting part of the argument is contained in the proof of the following lemma.

Lemma 48.34.1. Let $U$ be a variety. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. If $H^ d(U, \mathcal{F})$ is nonzero, then $\dim (U) \geq d$ and if equality holds, then $U$ is proper.

Proof. By the Grothendieck's vanishing result in Cohomology, Proposition 20.20.7 we conclude that $\dim (U) \geq d$. Assume $\dim (U) = d$. Choose a compactification $U \to X$ such that $U$ is dense in $X$. (This is possible by More on Flatness, Theorem 38.33.8 and Lemma 38.32.2.) After replacing $X$ by its reduction we find that $X$ is a proper variety of dimension $d$ and we see that $U$ is proper if and only if $U = X$. Set $Z = X \setminus U$. We will show that $H^ d(U, \mathcal{F})$ is zero if $Z$ is nonempty.

Choose a coherent $\mathcal{O}_ X$-module $\mathcal{G}$ whose restriction to $U$ is $\mathcal{F}$, see Properties, Lemma 28.22.5. Let $\omega _ X^\bullet $ denote the dualizing complex of $X$ as in Section 48.27. Set $\omega _ U^\bullet = \omega _ X^\bullet |_ U$. Then $H^ d(U, \mathcal{F})$ is dual to

\[ H^{-d}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}, \omega _ U^\bullet )) \]

by Lemma 48.33.1. By Lemma 48.27.1 we see that the cohomology sheaves of $\omega _ X^\bullet $ vanish in degrees $< -d$ and $H^{-d}(\omega _ X^\bullet ) = \omega _ X$ is a coherent $\mathcal{O}_ X$-module which is $(S_2)$ and whose support is $X$. In particular, $\omega _ X$ is torsion free, see Divisors, Lemma 31.11.10. Thus we see that the cohomology sheaf

\[ H^{-d}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \omega _ X^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \omega _ X) \]

is torsion free, see Divisors, Lemma 31.11.12. Consequently this sheaf has no nonzero sections vanishing on any nonempty open of $X$ (those would be torsion sections). Thus it follows from Lemma 48.33.3 that $H^{-d}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}, \omega _ U^\bullet ))$ is zero, and hence $H^ d(U, \mathcal{F})$ is zero as desired. $\square$

Theorem 48.34.2. Let $X$ be a nonempty separated scheme of finite type over a field $k$. Let $d = \dim (X)$. The following are equivalent

  1. $H^ d(X, \mathcal{F}) = 0$ for all coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ on $X$,

  2. $H^ d(X, \mathcal{F}) = 0$ for all quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ on $X$, and

  3. no irreducible component $X' \subset X$ of dimension $d$ is proper over $k$.

Proof. Assume there exists an irreducible component $X' \subset X$ (which we view as an integral closed subscheme) which is proper and has dimension $d$. Let $\omega _{X'}$ be a dualizing module of $X'$ over $k$, see Lemma 48.27.1. Then $H^ d(X', \omega _{X'})$ is nonzero as it is dual to $H^0(X', \mathcal{O}_{X'})$ by the lemma. Hence we see that $H^ d(X, \omega _{X'}) = H^ d(X', \omega _{X'})$ is nonzero and we conclude that (1) does not hold. In this way we see that (1) implies (3).

Let us prove that (3) implies (1). Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module such that $H^ d(X, \mathcal{F})$ is nonzero. Choose a filtration

\[ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F} \]

as in Cohomology of Schemes, Lemma 30.12.3. We obtain exact sequences

\[ H^ d(X, \mathcal{F}_ i) \to H^ d(X, \mathcal{F}_{i + 1}) \to H^ d(X, \mathcal{F}_{i + 1}/\mathcal{F}_ i) \]

Thus for some $i \in \{ 1, \ldots , m\} $ we find that $H^ d(X, \mathcal{F}_{i + 1}/\mathcal{F}_ i)$ is nonzero. By our choice of the filtration this means that there exists an integral closed subscheme $Z \subset X$ and a nonzero coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ such that $H^ d(Z, \mathcal{I})$ is nonzero. By Lemma 48.34.1 we conclude $\dim (Z) = d$ and $Z$ is proper over $k$ contradicting (3). Hence (3) implies (1).

Finally, let us show that (1) and (2) are equivalent for any Noetherian scheme $X$. Namely, (2) trivially implies (1). On the other hand, assume (1) and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then we can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as the filtered colimit of its coherent submodules, see Properties, Lemma 28.22.3. Then we have $H^ d(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ d(X, \mathcal{F}_ i) = 0$ by Cohomology, Lemma 20.19.1. Thus (2) is true. $\square$

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