Exercise 111.12.5. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Show that there exists a canonical short exact sequence

such that the following are true

$\text{depth}(Q) \geq 1$,

$K$ is zero or $\text{Supp}(K) = \{ \mathfrak m\} $, and

$\text{length}_ R(K) < \infty $.

Hint: using the Noetherian property show that there exists a maximal submodule $K$ as in (2) and then show that $Q = M/K$ satisfies (1) and $K$ satisfies (3).

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