The Stacks project

Remark 10.102.10. If in Proposition 10.102.9 the equivalent conditions (1) and (2) are satisfied, then there exists a $j$ such that $I(\varphi _ i) = R$ if and only if $i \geq j$. As in the proof of the proposition, it suffices to see this when all the matrices have coefficients in the maximal ideal $\mathfrak m$ of $R$. In this case we see that $I(\varphi _ j) = R$ if and only if $\varphi _ j = 0$. But if $\varphi _ j = 0$, then we get arbitrarily long exact complexes $0 \to R^{n_ e} \to R^{n_{e-1}} \to \ldots \to R^{n_ j} \to 0 \to 0 \to \ldots \to 0$ and hence by the proposition we see that $I(\varphi _ i)$ for $i > j$ has to be $R$ (since otherwise it is a proper ideal of a Noetherian local ring containing arbitrary long regular sequences which is impossible).