Lagmental Vicfred

Minimal Generators Are a Basis Modulo the Maximal Ideal by Vicfred

tags: mathematics, local algebra, Nakayama lemma, regular local rings

The smallest number of generators of a finite local module is the dimension of its residue vector space. I want the notation, the mechanism, and the failure mode visible at the same time.

The data

In a local ring \((A,\mathfrak m)\), reduction modulo \(\mathfrak m\) turns finite-module questions into linear algebra over the residue field \(k=A/\mathfrak m\).

$$ k=A/\mathfrak m,\qquad\mu_A(M)=\min\{|S|:S\text{ generates }M\} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \mu_A(M)=\dim_k(M/\mathfrak mM) $$

Derivation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \left.\begin{aligned}\bar m_1,\ldots,\bar m_r&\text{ basis of }M/\mathfrak mM\\N&=Am_1+\cdots+Am_r\end{aligned}\right\}\Longrightarrow M/N=\mathfrak m(M/N)=0 $$

Invariant content

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad k=A/\mathfrak m,\qquad\mu_A(M)=\min\{|S|:S\text{ generates }M\},\\[5pt] \mathsf{C}\;&:\quad \mu_A(M)=\dim_k(M/\mathfrak mM). \end{aligned} $$

Scope

Finite generation is essential in Nakayama's lemma. Infinite modules can satisfy \(\mathfrak mM=M\) without vanishing.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mu_A(M)=\dim_k(M/\mathfrak mM) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Mon 14 November 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.