Lemma 15.94.1. In Situation 15.91.15. If $A$ is Noetherian, then the pro-objects $\{ K_ n^\bullet \}$ and $\{ A/(f_1^ n, \ldots , f_ r^ n)\}$ of $D(A)$ are isomorphic1.

Proof. We have an inverse system of distinguished triangles

$\tau _{\leq -1}K_ n^\bullet \to K_ n^\bullet \to A/(f_1^ m, \ldots , f_ r^ m) \to (\tau _{\leq -1}K_ n^\bullet )[1]$

See Derived Categories, Remark 13.12.4. By Derived Categories, Lemma 13.42.4 it suffices to show that the inverse system $\tau _{\leq -1}K_ n^\bullet$ is pro-zero. Recall that $K_ n^\bullet$ has nonzero terms only in degrees $i$ with $-r \leq i \leq 0$. Thus by Derived Categories, Lemma 13.42.3 it suffices to show that $H^ p(K_ n^\bullet )$ is pro-zero for $p \leq -1$. In other words, for every $n \in \mathbf{N}$ we have to show there exists an $m \geq n$ such that $H^ p(K_ m^\bullet ) \to H^ p(K_ n^\bullet )$ is zero. Since $A$ is Noetherian, we see that

$H^ p(K_ n^\bullet ) = \frac{\mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})}{\mathop{\mathrm{Im}}(K_ n^{p - 1} \to K_ n^ p)}$

is a finite $A$-module. Moreover, the map $K_ m^ p \to K_ n^ p$ is given by a diagonal matrix whose entries are in the ideal $(f_1^{m - n}, \ldots , f_ r^{m - n})$ as $p < 0$. Note that $H^ p(K_ n^\bullet )$ is annihilated by $J = (f_1^ n, \ldots , f_ r^ n)$, see Lemma 15.28.6. Now $(f_1^{m - n}, \ldots , f_ r^{m - n}) \subset J^ t$ for $m - n \geq tn$. Thus by Algebra, Lemma 10.51.2 (Artin-Rees) applied to the ideal $J$ and the module $M = K_ n^ p$ with submodule $N = \mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$ for $m$ large enough the image of $K_ m^ p \to K_ n^ p$ intersected with $\mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$ is contained in $J \mathop{\mathrm{Ker}}(K_ n^ p \to K_ n^{p + 1})$. For such $m$ we get the zero map. $\square$

[1] In particular, for every $n$ there exists an $m \geq n$ such that $K_ m^\bullet \to K_ n^\bullet$ factors through the map $K_ m^\bullet \to A/(f_1^ m, \ldots , f_ r^ m)$.

Comment #6235 by Owen on

I think what we want and have is that $J^t\supset (f_1^{m-n},\ldots,f_r^{m-n})$ for $m-n\geq tn$, because we want to apply Artin-Rees for $I=J$, $M=K_n^p$, and $N=\operatorname{Ker}(K_n^p\to K_n^{p+1})$ (with the notation of 10.51.2).

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