The Stacks project

Lemma 15.100.2. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$, $N$ be $A$-modules with $M$ finite. For each $p > 0$ there exists a $c \geq 0$ such that for $n \geq c$ the map $\mathop{\mathrm{Ext}}\nolimits _ A^ p(M, N) \to \mathop{\mathrm{Ext}}\nolimits _ A^ p(I^ nM, N)$ factors through $\mathop{\mathrm{Ext}}\nolimits ^ p_ A(I^ nM, I^{n - c}N) \to \mathop{\mathrm{Ext}}\nolimits _ A^ p(I^ nM, N)$.

Proof. For $p = 0$, if $\varphi : M \to N$ is an $A$-linear map, then $\varphi (\sum f_ i m_ i) = \sum f_ i \varphi (m_ i)$ for $f_ i \in A$ and $m_ i \in M$. Hence $\varphi $ induces a map $I^ nM \to I^ nN$ for all $n$ and the result is true with $c = 0$.

Choose a short exact sequence $0 \to K \to A^{\oplus t} \to M \to 0$. For each $n$ we pick a short exact sequence $0 \to L_ n \to A^{\oplus s_ n} \to I^ nM \to 0$. It is clear that we can construct a map of short exact sequences

\[ \xymatrix{ 0 \ar[r] & L_ n \ar[r] \ar[d] & A^{\oplus s_ n} \ar[r] \ar[d] & I^ nM \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K \ar[r] & A^{\oplus t} \ar[r] & M \ar[r] & 0 } \]

such that $A^{\oplus s_ n} \to A^{\oplus t}$ has image in $(I^ n)^{\oplus t}$. By Artin-Rees (Algebra, Lemma 10.51.2) there exists a $c \geq 0$ such that $L_ n \to K$ factors through $I^{n - c}K$ if $n \geq c$.

For $p = 1$ our choices above induce a solid commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ A(A^{\oplus s_ n}, N) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ A(L_ n, N) \ar[r] & \mathop{\mathrm{Ext}}\nolimits _ A^1(I^ nM, N) \ar[r] & 0 \\ \mathop{\mathrm{Hom}}\nolimits _ A((I^ n)^{\oplus t}, I^{n - c}N) \ar[r] \ar[u] & \mathop{\mathrm{Hom}}\nolimits _ A(K \cap (I^ n)^{\oplus t}, I^{n - c}N) \ar[r] \ar[u] & \mathop{\mathrm{Ext}}\nolimits _ A^1(I^ nM, I^{n - c}N) \ar[u] \\ \mathop{\mathrm{Hom}}\nolimits _ A(A^{\oplus t}, N) \ar[r] \ar[u] & \mathop{\mathrm{Hom}}\nolimits _ A(K, N) \ar[r] \ar[u] & \mathop{\mathrm{Ext}}\nolimits _ A^1(M, N) \ar@{..>}[u] \ar[r] & 0 } \]

whose horizontal arrows are exact. The lower middle vertical arrow arises because $K \cap (I^ n)^{\oplus t} \subset I^{n - c}K$ and hence any $A$-linear map $K \to N$ induces an $A$-linear map $(I^ n)^{\oplus t} \to I^{n - c}N$ by the argument of the first paragraph. Thus we obtain the dotted arrow as desired.

For $p > 1$ we obtain a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(I^{n - c}K, N) \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(L_ n, N) \ar[r] & \mathop{\mathrm{Ext}}\nolimits _ A^ p(I^ nM, N) \\ \mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(K, N) \ar[rr] \ar[u] & & \mathop{\mathrm{Ext}}\nolimits _ A^ p(M, N) \ar[u] } \]

whose bottom horizontal arrow is an isomorphism. By induction on $p$ the left vertical map factors through $\mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(I^{n - c}K, I^{n - c - c'}N)$ for some $c' \geq 0$ and all $n \geq c + c'$. Using the composition $\mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(I^{n - c}K, I^{n - c - c'}N) \to \mathop{\mathrm{Ext}}\nolimits ^{p - 1}_ A(L_ n, I^{n - c - c'}N) \to \mathop{\mathrm{Ext}}\nolimits ^ p_ A(I^ nM, I^{n - c - c'}N)$ we obtain the desired factorization (for $n \geq c + c'$ and with $c$ replaced by $c + c'$). $\square$


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