A comparison of length definitions for maps of modules over local rings (Q1895177)

Let \(M\) be a finitely generated free module over a local ring \(R\) with maximal right ideal \(I\). Then \(\overline R = R/I\) is a skewfield. Put \(\overline M = M/MI\). An element \(\alpha\) in \(\Aut M\) is called simple if \(\text{codim} \overline {F(\alpha)} = 1\), where \(F (\alpha)\) is the set of elements in \(M\) which are fixed under the action of \(\alpha\). The element \(\alpha\) is called \(v\)-simple if \(\alpha = 1 + vf\), where \(v \in M \backslash \{0\}\) and \(f\) is in the dual of \(M\). Now \(\alpha\) is a product of simple \((v\)-simple) mappings. The minimal number of simple \((v\)-simple) mappings involved in any such factorization is the length \((v\)-length) of \(\alpha\), denoted by \(\ell (\alpha)\) \((v \ell (\alpha))\). Let \(B (\alpha) = (\alpha - 1)M\). Then \(\dim \overline {B(\alpha)} \leq v \ell (\alpha) \leq \ell (\alpha) \leq \text{codim} \overline {F(\alpha)}\). The author investigates the relation between \(\ell (\alpha)\) and \(v \ell (\alpha)\). She finds conditions for equalities and constructs examples where in the preceding string of inequalities no equality occurs. In order to do that, she introduces the tier\((k)\) condition.