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Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are linearly independent?

We can do this in the regular case, by just choosing a basis for the cotangent space and looking at their preimages in $m$ under the canonical map. By Nakayama's lemma these generate $m$ and hence form a system of parameters. Can we do this for any general Noetherian, local ring?

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    A minimal set of generators for $\mathfrak{m}$ should do it. If they were linearly dependent in $\mathfrak{m}/\mathfrak{m}^2$, then a smaller subset would suffice to generate $\mathfrak{m}$ by Nakayama.2010-11-29

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By induction let's just worry about picking one parameter element $x$. We need $x$ to be in $m$, but outside all the minimal primes of $R$ and $m^2$. This smells exactly like Prime Avoidance (note that they don't have to be all primes!). Now replace $R$ by $R/(x)$ and repeat.