Lemma 15.100.8 (0EGY)—The Stacks project

Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018.

Lemma 15.100.8. Let $I$ be an ideal of the Noetherian ring $A$ . Let $M$ and $N$ be finite $A$ -modules. Write $A_ n = A/I^ n$ , $M_ n = M/I^ nM$ , and $N_ n = N/I^ nN$ . For every $i \geq 0$ the objects

$\{ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, N)/I^ n\mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, N)\} _{n \geq 1} \quad \text{and}\quad \{ \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ n, N_ n)\} _{n \geq 1}$

are isomorphic in the category $\mathcal{C}$ of Remark 15.100.6.

Proof. Choose a short exact sequence

$0 \to K \to A^{\oplus r} \to M \to 0$

and set $K_ n = K/I^ nK$ . For $n \geq 1$ define $K(n) = \mathop{\mathrm{Ker}}(A_ n^{\oplus r} \to M_ n)$ so that we have exact sequences

$0 \to K(n) \to A_ n^{\oplus r} \to M_ n \to 0$

and surjections $K_ n \to K(n)$ . In fact, by Lemma 15.100.1 there is a $c \geq 0$ and maps $K(n) \to K_ n/I^{n - c}K_ n$ which are “almost inverse”. Since $I^{n - c}K_ n \subset K_ n[I^ c]$ these maps which witness the fact that the systems $\{ K(n)\} _{n \geq 1}$ and $\{ K_ n\} _{n \geq 1}$ are isomorphic in $\mathcal{C}$ .

We claim the systems

$\{ \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(K(n), N_ n)\} _{n \geq 1} \quad \text{and}\quad \{ \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(K_ n, N_ n)\} _{n \geq 1}$

are isomorphic in the category $\mathcal{C}$ . Namely, the surjective maps $K_ n \to K(n)$ have kernels annihilated by $I^ c$ and therefore determine maps

$\mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(K(n), N_ n) \to \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(K_ n, N_ n)$

whose kernel and cokernel are annihilated by $I^ c$ . Hence the claim by Lemma 15.100.7.

For $i \geq 2$ we have isomorphisms

$\mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ A(K, N) = \mathop{\mathrm{Ext}}\nolimits ^ i_ A(M, N) \quad \text{and}\quad \mathop{\mathrm{Ext}}\nolimits ^{i - 1}_{A_ n}(K(n), N_ n) = \mathop{\mathrm{Ext}}\nolimits ^ i_{A_ n}(M_ n, N_ n)$

In this way we see that it suffices to prove the lemma for $i = 0, 1$ .

For $i = 0, 1$ we consider the commutative diagram

$\xymatrix{ 0 \ar[r] & \mathop{\mathrm{Hom}}\nolimits (M, N) \ar[r] \ar[dd] & N^{\oplus r} \ar[r]_-\varphi \ar[dd] & \mathop{\mathrm{Hom}}\nolimits (K, N) \ar[r] \ar[d] & \mathop{\mathrm{Ext}}\nolimits ^1(M, N) \ar[r] & 0 \\ & & & \mathop{\mathrm{Hom}}\nolimits (K_ n, N_ n) \\ 0 \ar[r] & \mathop{\mathrm{Hom}}\nolimits (M_ n, N_ n) \ar[r] & N_ n^{\oplus r} \ar[r] & \mathop{\mathrm{Hom}}\nolimits (K(n), N_ n) \ar[r] \ar[u] & \mathop{\mathrm{Ext}}\nolimits ^1(M_ n, N_ n) \ar[r] & 0 }$

By Lemma 15.100.4 we see that the kernel and cokernel of $\mathop{\mathrm{Hom}}\nolimits (M, N)/I^ n \mathop{\mathrm{Hom}}\nolimits (M, N) \to \mathop{\mathrm{Hom}}\nolimits (M_ n, N_ n)$ and $\mathop{\mathrm{Hom}}\nolimits (K, N)/I^ n \mathop{\mathrm{Hom}}\nolimits (K, N) \to \mathop{\mathrm{Hom}}\nolimits (K_ n, N_ n)$ and are $I^ c$ -torsion for some $c \geq 0$ independent of $n$ . Above we have seen the cokernel of the injective maps $\mathop{\mathrm{Hom}}\nolimits (K(n), N_ n) \to \mathop{\mathrm{Hom}}\nolimits (K_ n, N_ n)$ are annihilated by $I^ c$ after possibly increasing $c$ . For such a $c$ we obtain maps $\delta _ n : I^ c\mathop{\mathrm{Hom}}\nolimits (K, N)/I^ n\mathop{\mathrm{Hom}}\nolimits (K, N) \to \mathop{\mathrm{Hom}}\nolimits (K(n), N_ n)$ fitting into the diagram (precise formulation omitted). The kernel and cokernel of $\delta _ n$ are annihilated by $I^ c$ after possibly increasing $c$ since we know that the same thing is true for $\mathop{\mathrm{Hom}}\nolimits (K, N)/I^ n \mathop{\mathrm{Hom}}\nolimits (K, N) \to \mathop{\mathrm{Hom}}\nolimits (K_ n, N_ n)$ and $\mathop{\mathrm{Hom}}\nolimits (K(n), N_ n) \to \mathop{\mathrm{Hom}}\nolimits (K_ n, N_ n)$ . Then we can use commutativity of the solid diagram

$\xymatrix{ \varphi ^{-1}(I^ c\mathop{\mathrm{Hom}}\nolimits (K, N)) \ar[r]_-\varphi \ar[d] & I^ c\mathop{\mathrm{Hom}}\nolimits (K, N)/I^ n\mathop{\mathrm{Hom}}\nolimits (K, N) \ar[r] \ar[d]^{\delta _ n} & I^ c\mathop{\mathrm{Ext}}\nolimits ^1(M, N)/I^ n\mathop{\mathrm{Ext}}\nolimits ^1(M, N) \ar[r] \ar@{..>}[d] & 0 \\ N_ n^{\oplus r} \ar[r] & \mathop{\mathrm{Hom}}\nolimits (K(n), N_ n) \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^1(M_ n, N_ n) \ar[r] & 0 }$

to define the dotted arrow. A straightforward diagram chase (omitted) shows that the kernel and cokernel of the dotted arrow are annihilated buy $I^ c$ after possibly increasing $c$ one final time. $\square$

The Stacks project

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