The Stacks project

Lemma 10.98.3. Let $A$ be a Noetherian graded ring. Let $I \subset A_+$ be a homogeneous ideal. Let $(N_ n)$ be an inverse system of finite graded $A$-modules with $N_ n = N_{n + 1}/I^ n N_{n + 1}$. Then there is a finite graded $A$-module $N$ such that $N_ n = N/I^ nN$ as graded modules for all $n$.

Proof. Pick $r$ and homogeneous elements $x_{1, 1}, \ldots , x_{1, r} \in N_1$ of degrees $d_1, \ldots , d_ r$ generating $N_1$. Since the transition maps are surjective, we can pick a compatible system of homogeneous elements $x_{n, i} \in N_ n$ lifting $x_{1, i}$. By the graded Nakayama lemma (Lemma 10.56.1) we see that $N_ n$ is generated by the elements $x_{n, 1}, \ldots , x_{n, r}$ sitting in degrees $d_1, \ldots , d_ r$. Thus for $m \leq n$ we see that $N_ n \to N_ n/I^ m N_ n$ is an isomorphism in degrees $< \min (d_ i) + m$ (as $I^ mN_ n$ is zero in those degrees). Thus the inverse system of degree $d$ parts

\[ \ldots = N_{2 + d - \min (d_ i), d} = N_{1 + d - \min (d_ i), d} = N_{d - \min (d_ i), d} \to N_{-1 + d - \min (d_ i), d} \to \ldots \]

stabilizes as indicated. Let $N$ be the graded $A$-module whose $d$th graded part is this stabilization. In particular, we have the elements $x_ i = \mathop{\mathrm{lim}}\nolimits x_{n, i}$ in $N$. We claim the $x_ i$ generate $N$: any $x \in N_ d$ is a linear combination of $x_1, \ldots , x_ r$ because we can check this in $N_{d - \min (d_ i), d}$ where it holds as $x_{d - \min (d_ i), i}$ generate $N_{d - \min (d_ i)}$. Finally, the reader checks that the surjective map $N/I^ nN \to N_ n$ is an isomorphism by checking to see what happens in each degree as before. Details omitted. $\square$

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