The Stacks project

29.23 Coherent formal modules

As we do not yet have the theory of formal schemes to our disposal, we develop a bit of language that replaces the notion of a “coherent module on a Noetherian adic formal scheme”.

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. We will consider inverse systems $(\mathcal{F}_ n)$ of coherent $\mathcal{O}_ X$-modules such that

  1. $\mathcal{F}_ n$ is annihilated by $\mathcal{I}^ n$, and

  2. the transition maps induce isomorphisms $\mathcal{F}_{n + 1}/\mathcal{I}^ n\mathcal{F}_{n + 1} \to \mathcal{F}_ n$.

A morphism of such inverse systems is defined as usual. Let us denote the category of these inverse systems with $\textit{Coh}(X, \mathcal{I})$. We are going to proceed by proving a bunch of lemmas about objects in this category. In fact, most of the lemmas that follow are straightforward consequences of the following description of the category in the affine case.

Lemma 29.23.1. If $X = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a Noetherian ring and $\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal $I \subset A$, then $\textit{Coh}(X, \mathcal{I})$ is equivalent to the category of finite $A^\wedge $-modules where $A^\wedge $ is the completion of $A$ with respect to $I$.

Proof. Let $\text{Mod}^{fg}_{A, I}$ be the category of inverse systems $(M_ n)$ of finite $A$-modules satisfying: (1) $M_ n$ is annihilated by $I^ n$ and (2) $M_{n + 1}/I^ nM_{n + 1} = M_ n$. By the correspondence between coherent sheaves on $X$ and finite $A$-modules (Lemma 29.9.1) it suffices to show $\text{Mod}^{fg}_{A, I}$ is equivalent to the category of finite $A^\wedge $-modules. To see this it suffices to prove that given an object $(M_ n)$ of $\text{Mod}^{fg}_{A, I}$ the module

\[ M = \mathop{\mathrm{lim}}\nolimits M_ n \]

is a finite $A^\wedge $-module and that $M/I^ nM = M_ n$. As the transition maps are surjective, we see that $M \to M_1$ is surjective. Pick $x_1, \ldots , x_ t \in M$ which map to generators of $M_1$. This induces a map of systems $(A/I^ n)^{\oplus t} \to M_ n$. By Nakayama's lemma (Algebra, Lemma 10.19.1) these maps are surjective. Let $K_ n \subset (A/I^ n)^{\oplus t}$ be the kernel. Property (2) implies that $K_{n + 1} \to K_ n$ is surjective, in particular the system $(K_ n)$ satisfies the Mittag-Leffler condition. By Homology, Lemma 12.29.3 we obtain an exact sequence $0 \to K \to (A^\wedge )^{\oplus t} \to M \to 0$ with $K = \mathop{\mathrm{lim}}\nolimits K_ n$. Hence $M$ is a finite $A^\wedge $-module. As $K \to K_ n$ is surjective it follows that

\[ M/I^ nM = \mathop{\mathrm{Coker}}(K \to (A/I^ n)^{\oplus t}) = (A/I^ n)^{\oplus t}/K_ n = M_ n \]

as desired. $\square$

Lemma 29.23.2. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals.

  1. The category $\textit{Coh}(X, \mathcal{I})$ is abelian.

  2. For $U \subset X$ open the restriction functor $\textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(U, \mathcal{I}|_ U)$ is exact.

  3. Exactness in $\textit{Coh}(X, \mathcal{I})$ may be checked by restricting to the members of an open covering of $X$.

Proof. Let $\alpha =(\alpha _ n) : (\mathcal{F}_ n) \to (\mathcal{G}_ n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. The cokernel of $\alpha $ is the inverse system $(\mathop{\mathrm{Coker}}(\alpha _ n))$ (details omitted). To describe the kernel let

\[ \mathcal{K}'_{l, m} = \mathop{\mathrm{Im}}(\mathop{\mathrm{Ker}}(\alpha _ l) \to \mathcal{F}_ m) \]

for $l \geq m$. We claim:

  1. the inverse system $(\mathcal{K}'_{l, m})_{l \geq m}$ is eventually constant, say with value $\mathcal{K}'_ m$,

  2. the system $(\mathcal{K}'_ m/\mathcal{I}^ n\mathcal{K}'_ m)_{m \geq n}$ is eventually constant, say with value $\mathcal{K}_ n$,

  3. the system $(\mathcal{K}_ n)$ forms an object of $\textit{Coh}(X, \mathcal{I})$, and

  4. this object is the kernel of $\alpha $.

To see (a), (b), and (c) we may work affine locally, say $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{I}$ corresponds to the ideal $I \subset A$. By Lemma 29.23.1 $\alpha $ corresponds to a map $f : M \to N$ of finite $A^\wedge $-modules. Denote $K = \mathop{\mathrm{Ker}}(f)$. Note that $A^\wedge $ is a Noetherian ring (Algebra, Lemma 10.96.6). Choose an integer $c \geq 0$ such that $K \cap I^ n M \subset I^{n - c}K$ for $n \geq c$ (Algebra, Lemma 10.50.2) and which satisfies Algebra, Lemma 10.50.3 for the map $f$ and the ideal $I^\wedge = IA^\wedge $. Then $\mathcal{K}'_{l, m}$ corresponds to the $A$-module

\[ K'_{l, m} = \frac{a^{-1}(I^ lN) + I^ mM}{I^ mM} = \frac{K + I^{l - c}f^{-1}(I^ cN) + I^ mM}{I^ mM} = \frac{K + I^ mM}{I^ mM} \]

where the last equality holds if $l \geq m + c$. So $\mathcal{K}'_ m$ corresponds to the $A$-module $K/K \cap I^ mM$ and $\mathcal{K}'_ m/\mathcal{I}^ n\mathcal{K}'_ m$ corresponds to

\[ \frac{K}{K \cap I^ mM + I^ nK} = \frac{K}{I^ nK} \]

for $m \geq n + c$ by our choice of $c$ above. Hence $\mathcal{K}_ n$ corresponds to $K/I^ nK$.

We prove (d). It is clear from the description on affines above that the composition $(\mathcal{K}_ n) \to (\mathcal{F}_ n) \to (\mathcal{G}_ n)$ is zero. Let $\beta : (\mathcal{H}_ n) \to (\mathcal{F}_ n)$ be a morphism such that $\alpha \circ \beta = 0$. Then $\mathcal{H}_ l \to \mathcal{F}_ l$ maps into $\mathop{\mathrm{Ker}}(\alpha _ l)$. Since $\mathcal{H}_ m = \mathcal{H}_ l/\mathcal{I}^ m\mathcal{H}_ l$ for $l \geq m$ we obtain a system of maps $\mathcal{H}_ m \to \mathcal{K}'_{l, m}$. Thus a map $\mathcal{H}_ m \to \mathcal{K}_ m'$. Since $\mathcal{H}_ n = \mathcal{H}_ m/\mathcal{I}^ n\mathcal{H}_ m$ we obtain a system of maps $\mathcal{H}_ n \to \mathcal{K}'_ m/\mathcal{I}^ n\mathcal{K}'_ m$ and hence a map $\mathcal{H}_ n \to \mathcal{K}_ n$ as desired.

To finish the proof of (1) we still have to show that $\mathop{\mathrm{Coim}}= \mathop{\mathrm{Im}}$ in $\textit{Coh}(X, \mathcal{I})$. We have seen above that taking kernels and cokernels commutes, over affines, with the description of $\textit{Coh}(X, \mathcal{I})$ as a category of modules. Since $\mathop{\mathrm{Im}}= \mathop{\mathrm{Coim}}$ holds in the category of modules this gives $\mathop{\mathrm{Coim}}= \mathop{\mathrm{Im}}$ in $\textit{Coh}(X, \mathcal{I})$. Parts (2) and (3) of the lemma are immediate from our construction of kernels and cokernels. $\square$

Lemma 29.23.3. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. A map $(\mathcal{F}_ n) \to (\mathcal{G}_ n)$ is surjective in $\textit{Coh}(X, \mathcal{I})$ if and only if $\mathcal{F}_1 \to \mathcal{G}_1$ is surjective.

Proof. Omitted. Hint: Look on affine opens, use Lemma 29.23.1, and use Algebra, Lemma 10.19.1. $\square$

Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. There is a functor

29.23.3.1
\begin{equation} \label{coherent-equation-completion-functor} \textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \end{equation}

which associates to the coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the object $\mathcal{F}^\wedge = (\mathcal{F}/\mathcal{I}^ n\mathcal{F})$ of $\textit{Coh}(X, \mathcal{I})$.

Proof. It suffices to check this locally on $X$. Hence we may assume $X$ is affine, i.e., we have a situation as in Lemma 29.23.1. The functor is the functor $\text{Mod}^{fg}_ A \to \text{Mod}^{fg}_{A^\wedge }$ which associates to a finite $A$-module $M$ the completion $M^\wedge $. Thus the result follows from Algebra, Lemma 10.96.2. $\square$

Lemma 29.23.5. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$. Then

\[ \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ). \]

Proof. To prove this we may work affine locally on $X$. Hence we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{F}$, $\mathcal{G}$ given by finite $A$-module $M$ and $N$. Then $\mathcal{H}$ corresponds to the finite $A$-module $H = \mathop{\mathrm{Hom}}\nolimits _ A(M, N)$. The statement of the lemma becomes the statement

\[ H^\wedge = \mathop{\mathrm{Hom}}\nolimits _{A^\wedge }(M^\wedge , N^\wedge ) \]

via the equivalence of Lemma 29.23.1. By Algebra, Lemma 10.96.2 (used 3 times) we have

\[ H^\wedge = \mathop{\mathrm{Hom}}\nolimits _ A(M, N) \otimes _ A A^\wedge = \mathop{\mathrm{Hom}}\nolimits _{A^\wedge }(M \otimes _ A A^\wedge , N \otimes _ A A^\wedge ) = \mathop{\mathrm{Hom}}\nolimits _{A^\wedge }(M^\wedge , N^\wedge ) \]

where the second equality uses that $A^\wedge $ is flat over $A$ (see More on Algebra, Remark 15.62.21). The lemma follows. $\square$

Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. We say an object $(\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ is $\mathcal{I}$-power torsion or is annihilated by a power of $\mathcal{I}$ if there exists a $c \geq 1$ such that $\mathcal{F}_ n = \mathcal{F}_ c$ for all $n \geq c$. If this is the case we will say that $(\mathcal{F}_ n)$ is annihilated by $\mathcal{I}^ c$. If $X = \mathop{\mathrm{Spec}}(A)$ is affine, then, via the equivalence of Lemma 29.23.1, these objects corresponds exactly to the finite $A$-modules annihilated by a power of $I$ or by $I^ c$.

Lemma 29.23.6. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module. Let $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$.

  1. If $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge $ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

    1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

    2. $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

    3. $\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and

    4. $\alpha = a^\wedge \circ \beta $.

  2. If $\alpha : \mathcal{G}^\wedge \to (\mathcal{F}_ n)$ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

    1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

    2. $a : \mathcal{G} \to \mathcal{F}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

    3. $\beta : \mathcal{F}^\wedge \to (\mathcal{F}_ n)$ is an isomorphism, and

    4. $\alpha = \beta \circ a^\wedge $.

Proof. Proof of (1). The uniqueness implies it suffices to construct $(\mathcal{F}, a, \beta )$ Zariski locally on $X$. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{I}$ corresponds to the ideal $I \subset A$. In this situation Lemma 29.23.1 applies. Let $M'$ be the finite $A^\wedge $-module corresponding to $(\mathcal{F}_ n)$. Let $N$ be the finite $A$-module corresponding to $\mathcal{G}$. Then $\alpha $ corresponds to a map

\[ \varphi : M' \longrightarrow N^\wedge \]

whose kernel and cokernel are annihilated by $I^ t$ for some $t$. Recall that $N^\wedge = N \otimes _ A A^\wedge $ (Algebra, Lemma 10.96.1). By More on Algebra, Lemma 15.80.16 there is an $A$-module map $\psi : M \to N$ whose kernel and cokernel are $I$-power torsion and an isomorphism $M \otimes _ A A^\wedge = M'$ compatible with $\varphi $. As $N$ and $M'$ are finite modules, we conclude that $M$ is a finite $A$-module, see More on Algebra, Remark 15.80.19. Hence $M \otimes _ A A^\wedge = M^\wedge $. We omit the verification that the triple $(M, N \to M, M^\wedge \to M')$ so obtained is unique up to unique isomorphism.

The proof of (2) is exactly the same and we omit it. $\square$

Lemma 29.23.7. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Any object of $\textit{Coh}(X, \mathcal{I})$ which is annihilated by a power of $\mathcal{I}$ is in the essential image of (29.23.3.1). Moreover, if $\mathcal{F}$, $\mathcal{G}$ are in $\textit{Coh}(\mathcal{O}_ X)$ and either $\mathcal{F}$ or $\mathcal{G}$ is annihilated by a power of $\mathcal{I}$, then the maps

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, \mathcal{G}) \ar[d] & \mathop{\mathrm{Ext}}\nolimits _ X(\mathcal{F}, \mathcal{G}) \ar[d] \\ \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge , \mathcal{G}^\wedge ) & \mathop{\mathrm{Ext}}\nolimits _{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge , \mathcal{G}^\wedge ) } \]

are isomorphisms.

Proof. Suppose $(\mathcal{F}_ n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ which is annihilated by $\mathcal{I}^ c$ for some $c \geq 1$. Then $\mathcal{F}_ n \to \mathcal{F}_ c$ is an isomorphism for $n \geq c$. Hence if we set $\mathcal{F} = \mathcal{F}_ c$, then we see that $\mathcal{F}^\wedge \cong (\mathcal{F}_ n)$. This proves the first assertion.

Let $\mathcal{F}$, $\mathcal{G}$ be objects of $\textit{Coh}(\mathcal{O}_ X)$ such that either $\mathcal{F}$ or $\mathcal{G}$ is annihilated by $\mathcal{I}^ c$ for some $c \geq 1$. Then $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ is a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}^ c$. Hence we see that

\[ \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = H^0(X, \mathcal{H}) = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ). \]

see Lemma 29.23.5. This proves the statement on homomorphisms.

The notation $\mathop{\mathrm{Ext}}\nolimits $ refers to extensions as defined in Homology, Section 12.6. The injectivity of the map on $\mathop{\mathrm{Ext}}\nolimits $'s follows immediately from the bijectivity of the map on $\mathop{\mathrm{Hom}}\nolimits $'s. For surjectivity, assume $\mathcal{F}$ is annihilated by a power of $I$. Then part (1) of Lemma 29.23.6 shows that given an extension

\[ 0 \to \mathcal{G}^\wedge \to (\mathcal{E}_ n) \to \mathcal{F}^\wedge \to 0 \]

in $\textit{Coh}(U, I\mathcal{O}_ U)$ the morphism $\mathcal{G}^\wedge \to (\mathcal{E}_ n)$ is isomorphic to $\mathcal{G} \to \mathcal{E}^\wedge $ for some $\mathcal{G} \to \mathcal{E}$ in $\textit{Coh}(\mathcal{O}_ U)$. Similarly in the other case. $\square$

Lemma 29.23.8. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. If $(\mathcal{F}_ n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ then $\bigoplus \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n)$ is a finite type, graded, quasi-coherent $\bigoplus \mathcal{I}^ n/\mathcal{I}^{n + 1}$-module.

Proof. The question is local on $X$ hence we may assume $X$ is affine, i.e., we have a situation as in Lemma 29.23.1. In this case, if $(\mathcal{F}_ n)$ corresponds to the finite $A^\wedge $ module $M$, then $\bigoplus \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n)$ corresponds to $\bigoplus I^ nM/I^{n + 1}M$ which is clearly a finite module over $\bigoplus I^ n/I^{n + 1}$. $\square$

Lemma 29.23.9. Let $f : X \to Y$ be a morphism of Noetherian schemes. Let $\mathcal{J} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals and set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X$. Then there is a right exact functor

\[ f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]

which sends $(\mathcal{G}_ n)$ to $(f^*\mathcal{G}_ n)$. If $f$ is flat, then $f^*$ is an exact functor.

Proof. Since $f^* : \textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(\mathcal{O}_ X)$ is right exact we have

\[ f^*\mathcal{G}_ n = f^*(\mathcal{G}_{n + 1}/\mathcal{I}^ n\mathcal{G}_{n + 1}) = f^*\mathcal{G}_{n + 1}/f^{-1}\mathcal{I}^ nf^*\mathcal{G}_{n + 1} = f^*\mathcal{G}_{n + 1}/\mathcal{J}^ nf^*\mathcal{G}_{n + 1} \]

hence the pullback of a system is a system. The construction of cokernels in the proof of Lemma 29.23.2 shows that $f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})$ is always right exact. If $f$ is flat, then $f^* : \textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(\mathcal{O}_ X)$ is an exact functor. It follows from the construction of kernels in the proof of Lemma 29.23.2 that in this case $f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})$ also transforms kernels into kernels. $\square$

Lemma 29.23.10. Let $f : X' \to X$ be a morphism of Noetherian schemes. Let $Z \subset X$ be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic inverse image. Let $\mathcal{I} \subset \mathcal{O}_ X$, $\mathcal{I}' \subset \mathcal{O}_{X'}$ be the corresponding quasi-coherent sheaves of ideals. If $f$ is flat and the induced morphism $Z' \to Z$ is an isomorphism, then the pullback functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', \mathcal{I}')$ (Lemma 29.23.9) is an equivalence.

Proof. If $X$ and $X'$ are affine, then this follows immediately from More on Algebra, Lemma 15.80.3. To prove it in general we let $Z_ n \subset X$, $Z'_ n \subset X'$ be the $n$th infinitesimal neighbourhoods of $Z$, $Z'$. The induced morphism $Z_ n \to Z'_ n$ is a homeomorphism on underlying topological spaces. On the other hand, if $z' \in Z'$ maps to $z \in Z$, then the ring map $\mathcal{O}_{X, z} \to \mathcal{O}_{X', z'}$ is flat and induces an isomorphism $\mathcal{O}_{X, z}/\mathcal{I}_ z \to \mathcal{O}_{X', z'}/\mathcal{I}'_{z'}$. Hence it induces an isomorphism $\mathcal{O}_{X, z}/\mathcal{I}_ z^ n \to \mathcal{O}_{X', z'}/(\mathcal{I}'_{z'})^ n$ for all $n \geq 1$ for example by More on Algebra, Lemma 15.80.2. Thus $Z'_ n \to Z_ n$ is an isomorphism of schemes. Thus $f^*$ induces an equivalence between the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{I}^ n$ and the category of coherent $\mathcal{O}_{X'}$-modules annihilated by $(\mathcal{I}')^ n$, see Lemma 29.9.8. This clearly implies the lemma. $\square$

Lemma 29.23.11. Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. If $V(\mathcal{I}) = V(\mathcal{J})$ is the same closed subset of $X$, then $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(X, \mathcal{J})$ are equivalent.

Proof. First, assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Let $I, J \subset A$ be the ideals corresponding to $\mathcal{I}, \mathcal{J}$. Then $V(I) = V(J)$ implies we have $I^ c \subset J$ and $J^ d \subset I$ for some $c, d \geq 1$ by elementary properties of the Zariski topology (see Algebra, Section 10.16 and Lemma 10.31.5). Hence the $I$-adic and $J$-adic completions of $A$ agree, see Algebra, Lemma 10.95.9. Thus the equivalence follows from Lemma 29.23.1 in this case.

In general, using what we said above and the fact that $X$ is quasi-compact, to choose $c, d \geq 1$ such that $\mathcal{I}^ c \subset \mathcal{J}$ and $\mathcal{J}^ d \subset \mathcal{I}$. Then given an object $(\mathcal{F}_ n)$ in $\textit{Coh}(X, \mathcal{I})$ we claim that the inverse system

\[ (\mathcal{F}_{cn}/\mathcal{J}^ n\mathcal{F}_{cn}) \]

is in $\textit{Coh}(X, \mathcal{J})$. This may be checked on the members of an affine covering; we omit the details. In the same manner we can construct an object of $\textit{Coh}(X, \mathcal{I})$ starting with an object of $\textit{Coh}(X, \mathcal{J})$. We omit the verification that these constructions define mutually quasi-inverse functors. $\square$


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