Lemma 42.68.27. Let A be a Noetherian local ring. Let M be a finite A-module of dimension 1. Assume \varphi , \psi : M \to M are two injective A-module maps, and assume \varphi (\psi (M)) = \psi (\varphi (M)), for example if \varphi and \psi commute. Then \text{length}_ R(M/\varphi \psi M) < \infty and (M/\varphi \psi M, \varphi , \psi ) is an exact (2, 1)-periodic complex.
Proof. Let \mathfrak q be a minimal prime of the support of M. Then M_{\mathfrak q} is a finite length A_{\mathfrak q}-module, see Algebra, Lemma 10.62.3. Hence both \varphi and \psi induce isomorphisms M_{\mathfrak q} \to M_{\mathfrak q}. Thus the support of M/\varphi \psi M is \{ \mathfrak m_ A\} and hence it has finite length (see lemma cited above). Finally, the kernel of \varphi on M/\varphi \psi M is clearly \psi M/\varphi \psi M, and hence the kernel of \varphi is the image of \psi on M/\varphi \psi M. Similarly the other way since M/\varphi \psi M = M/\psi \varphi M by assumption. \square
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