The Stacks project

51.19 Additional structure on local cohomology

Here is a sample result.

Lemma 51.19.1. Let $A$ be a ring. Let $I \subset A$ be a finitely generated ideal. Set $Z = V(I)$. For each derivation $\theta : A \to A$ there exists a canonical additive operator $D$ on the local cohomology modules $H^ i_ Z(A)$ satisfying the Leibniz rule with respect to $\theta $.

Proof. Let $f_1, \ldots , f_ r$ be elements generating $I$. Recall that $R\Gamma _ Z(A)$ is computed by the complex

\[ A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r} \]

See Dualizing Complexes, Lemma 47.9.1. Since $\theta $ extends uniquely to an additive operator on any localization of $A$ satisfying the Leibniz rule with respect to $\theta $, the lemma is clear. $\square$

Lemma 51.19.2. Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^ p$ the Frobenius endomorphism. Let $I \subset A$ be a finitely generated ideal. Set $Z = V(I)$. There exists an isomorphism $R\Gamma _ Z(A) \otimes _{A, F}^\mathbf {L} A \cong R\Gamma _ Z(A)$.

Proof. Follows from Dualizing Complexes, Lemma 47.9.3 and the fact that $Z = V(f_1^ p, \ldots , f_ r^ p)$ if $I = (f_1, \ldots , f_ r)$. $\square$

Lemma 51.19.3. Let $A$ be a ring. Let $V \to \mathop{\mathrm{Spec}}(A)$ be quasi-compact, quasi-separated, and étale. For each derivation $\theta : A \to A$ there exists a canonical additive operator $D$ on $H^ i(V, \mathcal{O}_ V)$ satisfying the Leibniz rule with respect to $\theta $.

Proof. If $V$ is separated, then we can argue using an affine open covering $V = \bigcup _{j = 1, \ldots m} V_ j$. Namely, because $V$ is separated we may write $V_{j_0 \ldots j_ p} = \mathop{\mathrm{Spec}}(B_{j_0 \ldots j_ p})$. See Schemes, Lemma 26.21.7. Then we find that the $A$-module $H^ i(V, \mathcal{O}_ V)$ is the $i$th cohomology group of the Čech complex

\[ \prod B_{j_0} \to \prod B_{j_0j_1} \to \prod B_{j_0j_1j_2} \to \ldots \]

See Cohomology of Schemes, Lemma 30.2.6. Each $B = B_{j_0 \ldots j_ p}$ is an étale $A$-algebra. Hence $\Omega _ B = \Omega _ A \otimes _ A B$ and we conclude $\theta $ extends uniquely to a derivation $\theta _ B : B \to B$. These maps define an endomorphism of the Čech complex and define the desired operators on the cohomology groups.

In the general case we use a hypercovering of $V$ by affine opens, exactly as in the first part of the proof of Cohomology of Schemes, Lemma 30.7.3. We omit the details. $\square$

Remark 51.19.4. We can upgrade Lemmas 51.19.1 and 51.19.3 to include higher order differential operators. If we ever need this we will state and prove a precise lemma here.

Lemma 51.19.5. Let $p$ be a prime number. Let $A$ be a ring with $p = 0$. Denote $F : A \to A$, $a \mapsto a^ p$ the Frobenius endomorphism. If $V \to \mathop{\mathrm{Spec}}(A)$ is quasi-compact, quasi-separated, and étale, then there exists an isomorphism $R\Gamma (V, \mathcal{O}_ V) \otimes _{A, F}^\mathbf {L} A \cong R\Gamma (V, \mathcal{O}_ V)$.

Proof. Observe that the relative Frobenius morphism

\[ V \longrightarrow V \times _{\mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(F)} \mathop{\mathrm{Spec}}(A) \]

of $V$ over $A$ is an isomorphism, see Étale Morphisms, Lemma 41.14.3. Thus the lemma follows from cohomology and base change, see Derived Categories of Schemes, Lemma 36.21.5. Observe that since $V$ is étale over $A$, it is flat over $A$. $\square$


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