The Stacks project

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. $\square$

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

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