The Stacks project

Lemma 30.25.3. Let $Y$ be a Noetherian scheme. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_ Y$ be quasi-coherent sheaves of ideals. Let $f : X \to Y$ be a proper morphism which is an isomorphism over $V = Y \setminus V(\mathcal{K})$. Set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X$. Let $(\mathcal{G}_ n)$ be an object of $\textit{Coh}(Y, \mathcal{J})$, let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module, and let $\beta : (f^*\mathcal{G}_ n) \to \mathcal{F}^\wedge $ be an isomorphism in $\textit{Coh}(X, \mathcal{I})$. Then there exists a map

\[ \alpha : (\mathcal{G}_ n) \longrightarrow (f_*\mathcal{F})^\wedge \]

in $\textit{Coh}(Y, \mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.

Proof. Since $f$ is a proper morphism we see that $f_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module (Proposition 30.19.1). Thus the statement of the lemma makes sense. Consider the compositions

\[ \gamma _ n : \mathcal{G}_ n \to f_*f^*\mathcal{G}_ n \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}). \]

Here the first map is the adjunction map and the second is $f_*\beta _ n$. We claim that there exists a unique $\alpha $ as in the lemma such that the compositions

\[ \mathcal{G}_ n \xrightarrow {\alpha _ n} f_*\mathcal{F}/\mathcal{J}^ nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}) \]

equal $\gamma _ n$ for all $n$. Because of the uniqueness we may assume that $Y = \mathop{\mathrm{Spec}}(B)$ is affine. Let $J \subset B$ corresponds to the ideal $\mathcal{J}$. Set

\[ M_ n = H^0(X, \mathcal{F}/\mathcal{I}^ n\mathcal{F}) \quad \text{and}\quad M = H^0(X, \mathcal{F}) \]

By Lemma 30.20.4 and Theorem 30.20.5 the inverse limit of the modules $M_ n$ equals the completion $M^\wedge = \mathop{\mathrm{lim}}\nolimits M/J^ nM$. Set $N_ n = H^0(Y, \mathcal{G}_ n)$ and $N = \mathop{\mathrm{lim}}\nolimits N_ n$. Via the equivalence of categories of Lemma 30.23.1 the finite $B^\wedge $ modules $N$ and $M^\wedge $ correspond to $(\mathcal{G}_ n)$ and $f_*\mathcal{F}^\wedge $. It follows from this that $\alpha $ has to be the morphism of $\textit{Coh}(Y, \mathcal{J})$ corresponding to the homomorphism

\[ \mathop{\mathrm{lim}}\nolimits \gamma _ n : N = \mathop{\mathrm{lim}}\nolimits _ n N_ n \longrightarrow \mathop{\mathrm{lim}}\nolimits M_ n = M^\wedge \]

of finite $B^\wedge $-modules.

We still have to show that the kernel and cokernel of $\alpha $ are annihilated by a power of $\mathcal{K}$. Set $Y' = \mathop{\mathrm{Spec}}(B^\wedge )$ and $X' = Y' \times _ Y X$. Let $\mathcal{K}'$, $\mathcal{J}'$, $\mathcal{G}'_ n$ and $\mathcal{I}'$, $\mathcal{F}'$ be the pullback of $\mathcal{K}$, $\mathcal{J}$, $\mathcal{G}_ n$ and $\mathcal{I}$, $\mathcal{F}$, to $Y'$ and $X'$. The projection morphism $f' : X' \to Y'$ is the base change of $f$ by $Y' \to Y$. Note that $Y' \to Y$ is a flat morphism of schemes as $B \to B^\wedge $ is flat by Algebra, Lemma 10.97.2. Hence $f'_*\mathcal{F}'$, resp. $f'_*(f')^*\mathcal{G}_ n'$ is the pullback of $f_*\mathcal{F}$, resp. $f_*f^*\mathcal{G}_ n$ to $Y'$ by Lemma 30.5.2. The uniqueness of our construction shows the pullback of $\alpha $ to $Y'$ is the corresponding map $\alpha '$ constructed for the situation on $Y'$. Moreover, to check that the kernel and cokernel of $\alpha $ are annihilated by $\mathcal{K}^ t$ it suffices to check that the kernel and cokernel of $\alpha '$ are annihilated by $(\mathcal{K}')^ t$. Namely, to see this we need to check this for kernels and cokernels of the maps $\alpha _ n$ and $\alpha '_ n$ (see Remark 30.25.1) and the ring map $B \to B^\wedge $ induces an equivalence of categories between modules annihilated by $J^ n$ and $(J')^ n$, see More on Algebra, Lemma 15.89.3. Thus we may assume $B$ is complete with respect to $J$.

Assume $Y = \mathop{\mathrm{Spec}}(B)$ is affine, $\mathcal{J}$ corresponds to the ideal $J \subset B$, and $B$ is complete with respect to $J$. In this case $(\mathcal{G}_ n)$ is in the essential image of the functor $\textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(Y, \mathcal{J})$. Say $\mathcal{G}$ is a coherent $\mathcal{O}_ Y$-module such that $(\mathcal{G}_ n) = \mathcal{G}^\wedge $. Note that $f^*(\mathcal{G}^\wedge ) = (f^*\mathcal{G})^\wedge $. Hence Lemma 30.24.1 tells us that $\beta $ comes from an isomorphism $b : f^*\mathcal{G} \to \mathcal{F}$ and $\alpha $ is the completion functor applied to

\[ \mathcal{G} \to f_*f^*\mathcal{G} \cong f_*\mathcal{F} \]

Hence we are trying to verify that the kernel and cokernel of the adjunction map $c : \mathcal{G} \to f_*f^*\mathcal{G}$ are annihilated by a power of $\mathcal{K}$. However, since the restriction $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is an isomorphism we see that $c|_ V$ is an isomorphism. Thus the coherent sheaves $\mathop{\mathrm{Ker}}(c)$ and $\mathop{\mathrm{Coker}}(c)$ are supported on $V(\mathcal{K})$ hence are annihilated by a power of $\mathcal{K}$ (Lemma 30.10.2) as desired. $\square$

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