= Minimal number of generators of a module
{title2=$\mu_A(M)=\dim_{A/\mathfrak m}(M/\mathfrak mM)\quad(M\text{ finite},\ A\text{ local})$}
For a <module> $M$, $\mu_A(M)$ is the least cardinality of a generating set, equivalently the least size of a <basis> of a <free module> surjecting onto $M$. If $(A,\mathfrak m)$ is a <local ring> and $M$ is finitely generated, then $\mu_A(M)=\dim_{A/\mathfrak m}(M/\mathfrak mM)$. Any generating set spans this quotient, proving the lower bound. Lift a <basis> of the quotient and let $J$ be the <submodule> generated by those lifts. Then $N=M/J$ is finitely generated and satisfies $N=\mathfrak mN$. For generators $n_i$ write $n_i=\sum_jc_{ij}n_j$ with $c_{ij}\in\mathfrak m$. The adjugate of $I-C$ shows that $\det(I-C)$ annihilates $N$; this determinant is one modulo $\mathfrak m$ and hence a <unit>. Thus $N=0$, giving the upper bound.
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