Let $X\rightarrow \operatorname{Spec}A$ be a morphism of schemes, $I$ an ideal of $A$ and $\mathcal{F}$ an $O_X-$module. The morphism gives a morphism $A\rightarrow O_X(X)$ and so for any open subset $U\subset X$, we've got a morphism $A\rightarrow O_X(U)$ and so $\mathcal{F}(U)$ can be equipped with an $A-$module structure and so we can consider the $O_X-$module $I\mathcal{F}:U\mapsto I\mathcal{F}(U)$ which is a submodule of $\mathcal{F}$.
I want to show that $IH^n(X,\mathcal{F}/I\mathcal{F})$ where $H^n$ denotes sheaf cohomology. I'd like to check whether my answer is correct, and I have a feeling I'm complicating things so if anyone has an easier solution I would appreciate it!

For any $x\in X$, $(\mathcal{F}/I\mathcal{F})_x$ is an $O_{X,x}-$module and as an $A-$module it is annihilated by $I$ so $(\mathcal{F}/I\mathcal{F})_x$ is an $A/I-$module.
Let $(\mathcal{F}/I\mathcal{F})_x\rightarrow J_x$ be a monomorphism to an injective $A/I-$module. Now we consider the sheaf of abelian groups $\mathcal{J}$ defined by $\mathcal{J}(U)=\prod_{x\in U}J_x.$ We can check that this is an injective sheaf of abelian groups and that the canonical morphism $\mathcal{F}/I\mathcal{F}\rightarrow \mathcal{J}$ is injective. Each $\mathcal{J}(U)$ is an $A/I-$module so we can construct an injective resolution $(\mathcal{J}^k)$ of $\mathcal{F}/I\mathcal{F}$ in the category of sheaves of abelian groups such that $\mathcal{J}^k(U)$ is an $A/I-$module for all $k$ and $U\subset X.$ So the cohomology $H^n(X,\mathcal{F}/I\mathcal{F})$ of the complex $(\mathcal{J}^k(X))$ is an $A/I-$module so $IH^n(X,\mathcal{F}/I\mathcal{F})=0.$

Thank you for your answers!


1 Answer 1


This might be equivalent to what you did if you go through all the proofs (and indeed you have understood the point, which is that an $A/I$-module is just an $A$-module killed by $I$). But it might still be helpful for you. Let us write down the setup in fancier symbols. We have a morphism of schemes $f: X \to \mathrm{Spec} A$, and a quasicoherent sheaf $\mathcal{F}$ on $X$. We also have a closed embedding $\iota: \mathrm{Spec}(A/I) \to \mathrm{Spec} A$. Let $$\iota': X \times_A A/I \to X$$ be the closed embedding induced by base-change to $X$ via $f$ (this is basically the preimage in $X$ of the closed subscheme cut out by $I$). Note that $\iota'$ is an affine morphism. The definition of the sheaf $\mathcal{F}/I\mathcal{F}$ considered as a quasicoherent sheaf on $X$ is $$\mathcal{F}/I\mathcal{F} := (\iota')_*(\iota')^*\mathcal{F}$$ (the pullback quotients by $I$ and the pushforward restricts scalars from $A/I$ to $A$ in order to consider what is naturally a sheaf on the closed subscheme $X \times_A A/I$ as a sheaf on all of $X$). Now we may apply the fact that $$H^n(X, \mathcal{F}/I\mathcal{F}) = H^n(X, (\iota')_*(\iota')^*\mathcal{F}) \cong H^n(X \times_A A/I, (\iota')^*\mathcal{F})$$ as $A$-modules (use e.g. Vakil 18.1(v)). In particular, the guy on the right is an $A/I$-module (which is where the $A$-module structure comes from), so we conclude that the $A$-module on the left is killed by $I$, as desired.


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