The Stacks project

Lemma 15.27.4. Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $(M_ n)$ be an inverse system of $A$-modules such that

  1. $M_ n$ is a flat $A/I^ n$-module,

  2. $M_{n + 1} \to M_ n$ is surjective.

Then $M = \mathop{\mathrm{lim}}\nolimits M_ n$ is a flat $A$-module and $Q \otimes _ A M = \mathop{\mathrm{lim}}\nolimits Q \otimes _ A M_ n$ for every finite $A$-module $Q$.

Proof. We first show that $Q \otimes _ A M = \mathop{\mathrm{lim}}\nolimits Q \otimes _ A M_ n$ for every finite $A$-module $Q$. Choose a resolution $F_2 \to F_1 \to F_0 \to Q \to 0$ by finite free $A$-modules $F_ i$. Then

\[ F_2 \otimes _ A M_ n \to F_1 \otimes _ A M_ n \to F_0 \otimes _ A M_ n \]

is a chain complex whose homology in degree $0$ is $Q \otimes _ A M_ n$ and whose homology in degree $1$ is

\[ \text{Tor}_1^ A(Q, M_ n) = \text{Tor}_1^ A(Q, A/I^ n) \otimes _{A/I^ n} M_ n \]

as $M_ n$ is flat over $A/I^ n$. By Lemma 15.27.3 we see that this system is essentially constant (with value $0$). It follows from Homology, Lemma 12.31.7 that $\mathop{\mathrm{lim}}\nolimits Q \otimes _ A A/I^ n = \mathop{\mathrm{Coker}}(\mathop{\mathrm{lim}}\nolimits F_1 \otimes _ A M_ n \to \mathop{\mathrm{lim}}\nolimits F_0 \otimes _ A M_ n)$. Since $F_ i$ is finite free this equals $\mathop{\mathrm{Coker}}(F_1 \otimes _ A M \to F_0 \otimes _ A M) = Q \otimes _ A M$.

Next, let $Q \to Q'$ be an injective map of finite $A$-modules. We have to show that $Q \otimes _ A M \to Q' \otimes _ A M$ is injective (Algebra, Lemma 10.38.5). By the above we see

\[ \mathop{\mathrm{Ker}}(Q \otimes _ A M \to Q' \otimes _ A M) = \mathop{\mathrm{Ker}}(\mathop{\mathrm{lim}}\nolimits Q \otimes _ A M_ n \to \mathop{\mathrm{lim}}\nolimits Q' \otimes _ A M_ n). \]

For each $n$ we have an exact sequence

\[ \text{Tor}_1^ A(Q', M_ n) \to \text{Tor}_1^ A(Q'', M_ n) \to Q \otimes _ A M_ n \to Q' \otimes _ A M_ n \]

where $Q'' = \mathop{\mathrm{Coker}}(Q \to Q')$. Above we have seen that the inverse systems of Tor's are essentially constant with value $0$. It follows from Homology, Lemma 12.31.7 that the inverse limit of the right most maps is injective. $\square$


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