The Stacks project

Proof. We have $H_ n = \mathop{\mathrm{Ker}}(\beta ) \cap I^ nL/\alpha (I^ nK)$. Since $\mathop{\mathrm{Ker}}(\beta ) \cap I^ nL \subset \mathop{\mathrm{Ker}}(\beta ) \cap I^{n - 1}L$ and $\alpha (I^ nK) \subset \alpha (I^{n - 1}K)$ we get the maps $H_ n \to H_{n - 1}$. Similarly for the map $H_1 \to H$.

By Artin-Rees we may choose $c_1, c_2 \geq 0$ such that $\mathop{\mathrm{Im}}(\alpha ) \cap I^ nL \subset \alpha (I^{n - c_1}K)$ for $n \geq c_1$ and $\mathop{\mathrm{Ker}}(\beta ) \cap I^ nL \subset I^{n - c_2}\mathop{\mathrm{Ker}}(\beta )$ for $n \geq c_2$, see Algebra, Lemmas 10.51.3 and 10.51.2. Set $c = c_1 + c_2$.

It follows immediately from our choice of $c \geq c_2$ that for $n \geq c$ the image of $H_ n \to H$ is contained in $I^{n - c}H$.

Let $n \geq c$. We define $\psi _ n : I^ nH \to H_{n - c}$ as follows. Say $x \in I^ nH$. Choose $y \in I^ n\mathop{\mathrm{Ker}}(\beta )$ representing $x$. We set $\psi _ n(x)$ equal to the class of $y$ in $H_{n - c}$. To see this is well defined, suppose we have a second choice $y'$ as above for $x$. Then $y' - y \in \mathop{\mathrm{Im}}(\alpha )$. By our choice of $c \geq c_1$ we conclude that $y' - y \in \alpha (I^{n - c}K)$ which implies that $y$ and $y'$ represent the same element of $H_{n - c}$. Thus $\psi _ n$ is well defined.

The statements on the compositions $I^ n H \to H_{n - c} \to I^{n - 2c}H$ and $H_ n \to I^{n - c}H \to H_{n - 2c}$ follow immediately from our definitions. $\square$