## 38.42 An operator introduced by Berthelot and Ogus

This section continuous the discussion started in More on Algebra, Section 15.94. We encourage the reader to read that section first.

Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. If $\mathcal{F}$ is an $\mathcal{O}_ X$-module1, then the following are equivalent

1. the subsheaf $\mathcal{F}[\mathcal{I}] \subset \mathcal{F}$ of sections annihilated by $\mathcal{I}$ (compare with Properties, Definition 28.24.3) is zero,

2. the multiplication map $\mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}$ is injective,

3. for every affine open $U = \mathop{\mathrm{Spec}}(A)$ such that $D \cap U = V(f)$ for a nonzerodivisor $f \in A$ (Divisors, Lemma 31.13.2), the map $f : \mathcal{F}|_ U \to \mathcal{F}|_ U$ is injective,

4. for every $x \in D$ and generator $f$ of the ideal $\mathcal{I}_ x \subset \mathcal{O}_{X, x}$ the element $f$ is a nonzerodivisor on the stalk $\mathcal{F}_ x$.

If these equivalent conditions hold, then we will say that $\mathcal{F}$ is $\mathcal{I}$-torsion free. If so, then for any $i \in \mathbf{Z}$ we will denote

$\mathcal{I}^ i\mathcal{F} = \mathcal{I}^{\otimes i} \otimes _{\mathcal{O}_ X} \mathcal{F} = \mathcal{O}_ X(-iD) \otimes _{\mathcal{O}_ X} \mathcal{F} = \mathcal{F}(-iD)$

so that we have inclusions

$\ldots \subset \mathcal{I}^{i + 1}\mathcal{F} \subset \mathcal{I}^ i\mathcal{F} \subset \mathcal{I}^{i - 1}\mathcal{F} \subset \ldots$

The modules $\mathcal{I}^ i\mathcal{F}$ are locally isomorphic to $\mathcal{F}$ as $\mathcal{O}_ X$-modules, but not globally.

Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_ X$-modules with differentials $d^ i : \mathcal{F}^ i \to \mathcal{F}^{i + 1}$ and assume $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. In this case we define $\eta _\mathcal {I}\mathcal{F}^\bullet$ to be the complex with terms

\begin{align*} (\eta _\mathcal {I}\mathcal{F})^ i & = \mathop{\mathrm{Ker}}\left( d^ i, -1 : \mathcal{I}^ i\mathcal{F}^ i \oplus \mathcal{I}^{i + 1}\mathcal{F}^{i + 1} \to \mathcal{I}^ i\mathcal{F}^{i + 1} \right) \\ & = \mathop{\mathrm{Ker}}\left(d^ i : \mathcal{I}^ i\mathcal{F}^ i \to \mathcal{I}^ i\mathcal{F}^{i + 1}/ \mathcal{I}^{i + 1}\mathcal{F}^{i + 1} \right) \end{align*}

and differential induced by $d^ i$. In other words, a local section $s$ of $(\eta _\mathcal {I}\mathcal{F})^ i$ is the same thing as a local section $s$ of $\mathcal{I}^ i\mathcal{F}^ i$ such that its image $d^ i(s)$ in $\mathcal{I}^ i\mathcal{F}^{i + 1}$ is in the subsheaf $\mathcal{I}^{i + 1}\mathcal{F}^{i + 1}$. Observe that $\eta _\mathcal {I}\mathcal{F}^\bullet$ is another complex whose terms are $\mathcal{I}$-torsion free modules.

Lemma 38.42.1. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$.

1. For $x \in X$ choose a generator $f \in \mathcal{I}_ x$. Then the stalk of $\eta _\mathcal {I}\mathcal{F}^\bullet$ is canonically isomorphic to $\eta _ f\mathcal{F}^\bullet _ x$.

2. If the $\mathcal{F}^ i$ are quasi-coherent $\mathcal{O}_ X$-modules, then so are the $(\eta _\mathcal {I}\mathcal{F})^ i$ and in this case if $U = \mathop{\mathrm{Spec}}(A) \subset X$ is affine open and $D \cap U = V(f)$, then $\eta _ f(\mathcal{F}^\bullet (U))$ is canonically isomorphic to $(\eta _\mathcal {I}\mathcal{F}^\bullet )(U)$.

Proof. Omitted. $\square$

Lemma 38.42.2. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. There is a canonical isomorphism

$\mathcal{I}^{\otimes i} \otimes _{\mathcal{O}_ X} \left( H^ i(\mathcal{F}^\bullet )/H^ i(\mathcal{F}^\bullet )[\mathcal{I}] \right) \longrightarrow H^ i(\eta _\mathcal {I}\mathcal{F}^\bullet )$

of cohomology sheaves.

Proof. Via Lemma 38.42.1 this translates into the result of More on Algebra, Lemma 15.94.1. $\square$

Lemma 38.42.3. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ be a map of complexes of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ and $\mathcal{G}^ i$ are $\mathcal{I}$-torsion free for all $i$. Then the induced map $\eta _\mathcal {I}\mathcal{F}^\bullet \to \eta _\mathcal {I}\mathcal{G}^\bullet$ is a quasi-isomorphism too.

Proof. This is true because the isomorphisms of Lemma 38.42.2 are compatible with maps of complexes. $\square$

Remark 38.42.4. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{G}^\bullet$ be a complex of $\mathcal{O}_ X$-modules. By Cohomology, Lemma 20.26.12 there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ such that $\mathcal{F}^\bullet$ is a K-flat complex whose terms are flat $\mathcal{O}_ X$-modules. (Even if $\mathcal{G}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_ X$-modules, in general $\mathcal{F}^\bullet$ will not be so.) It follows that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. In this situation we define

$L\eta _\mathcal {I} \mathcal{G}^\bullet = \eta _\mathcal {I} \mathcal{F}^\bullet$

This is independent of the choice of the K-flat resolution by Lemma 38.42.3. We obtain a functor $L\eta _\mathcal {I} : D(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$. Beware that this functor isn't exact, i.e., does not tranform distinguished triangles into distinguished triangles.

Lemma 38.42.5. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then $\eta _\mathcal {I}(\mathcal{F}^\bullet \otimes \mathcal{L}) = (\eta _\mathcal {I}\mathcal{F}^\bullet ) \otimes \mathcal{L}$.

Proof. Immediate from the construction. $\square$

 In this section we work with $\mathcal{O}_ X$-modules which are not necessarily quasi-coherent.

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