The Stacks project

Lemma 52.4.4. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi _ j : I^ c \to A$, $j = 1, \ldots , r$ be as in Lemma 52.4.1. Let $X$ be a Noetherian scheme over $\mathop{\mathrm{Spec}}(A)$. Let

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

be an inverse system of coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. Then, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, there exists a unique graded $\mathcal{O}_ X$-module map

\[ \Phi _\mathcal {F} : \bigoplus \nolimits _{n \geq 0} I^{nc}\mathcal{F} \longrightarrow \mathcal{F}[T_1, \ldots , T_ r] \]

with $\Phi _\mathcal {F}(g s) = \Phi (g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$ where $\Phi $ is as in Lemma 52.4.2. The composition of $\Phi _\mathcal {F}$ with the map $\mathcal{F}[T_1, \ldots , T_ r] \to \bigoplus _{n \geq 0} I^ n\mathcal{F}$, $T_ j \mapsto f_ j$ is the canonical inclusion $\bigoplus _{n \geq 0} I^{nc}\mathcal{F} \to \bigoplus _{n \geq 0} I^ n\mathcal{F}$.

Proof. The uniqueness is immediate from the $\mathcal{O}_ X$-linearity and the requirement that $\Phi _\mathcal {F}(g s) = \Phi (g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$. Thus we may assume $X = \mathop{\mathrm{Spec}}(B)$ is affine. Observe that $(\mathcal{F}_ n)$ is an object of the category $\textit{Coh}(X, I\mathcal{O}_ X)$ introduced in Cohomology of Schemes, Section 30.23. Let $B' = B^\wedge $ be the $I$-adic completion of $B$. By Cohomology of Schemes, Lemma 30.23.1 the object $(\mathcal{F}_ n)$ corresponds to a finite $B'$-module $N$ in the sense that $\mathcal{F}_ n$ is the coherent module associated to the finite $B$-module $N/I^ n N$. Applying Lemma 52.4.3 to $I \subset A \to B'$ and $N$ we see that, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, we get unique maps

\[ \Phi _ N : \bigoplus \nolimits _{n \geq 0} I^{nc}N \to N[T_1, \ldots , T_ r] \]

with the corresponding properties. Note that in degree $n$ we obtain an inverse system of maps $N/I^ mN \to \bigoplus _{e_1 + \ldots + e_ r = n} N/I^{m - nc}N \cdot T_1^{e_1} \ldots T_ r^{e_ r}$ for $m \geq nc$. Translating back into coherent sheaves we see that $\Phi _ N$ corresponds to a system of maps

\[ \Phi ^ n_ m : I^{nc}\mathcal{F}_ m \longrightarrow \bigoplus \nolimits _{e_1 + \ldots + e_ r = n} \mathcal{F}_{m - nc} \cdot T_1^{e_1} \ldots T_ r^{e_ r} \]

for varying $m \geq nc$ and $n \geq 1$. Taking the inverse limit of these maps over $m$ we obtain $\Phi _\mathcal {F} = \bigoplus _ n \mathop{\mathrm{lim}}\nolimits _ m \Phi ^ n_ m$. Note that $\mathop{\mathrm{lim}}\nolimits _ m I^ t\mathcal{F}_ m = I^ t \mathcal{F}$ as can be seen by evaluating on affines for example, but in fact we don't need this because it is clear there is a map $I^ t\mathcal{F} \to \mathop{\mathrm{lim}}\nolimits _ m I^ t\mathcal{F}_ m$. $\square$

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