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. $\square$

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.

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 0EGY. Beware of the difference between the letter 'O' and the digit '0'.