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 $I^ t\mathcal{F} \to \mathop{\mathrm{lim}}\nolimits _ m I^ t\mathcal{F}_ m$. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EH6. Beware of the difference between the letter 'O' and the digit '0'.