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 .
51.19 Additional structure on local cohomology
Here is a sample result.
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 ith 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 36.22.5. Observe that since V is étale over A, it is flat over A. \square
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