Forster–Swan theorem

Let

• ${\displaystyle R}$ be a commutative Noetherian ring with one,
• ${\displaystyle M}$ be a finitely generated ${\displaystyle R}$-module,
• ${\displaystyle {\mathfrak {p}}}$ a prime ideal of ${\displaystyle R}$.
• ${\displaystyle \mu (M)}$ and ${\displaystyle \mu _{\mathfrak {p}}(M)}$ are the minimal number of generators needed to generate the ${\displaystyle R}$-module ${\displaystyle M}$ and the ${\displaystyle R_{\mathfrak {p}}}$-module ${\displaystyle M_{\mathfrak {p}}}$, respectively.

According to Nakayama's lemma, in order to compute ${\displaystyle \mu _{\mathfrak {p}}(M)}$ one can compute the dimension of ${\displaystyle M_{\mathfrak {p}}/{\mathfrak {p}}M}$ over the field ${\displaystyle k({\mathfrak {p}})=R_{\mathfrak {p}}/{\mathfrak {p}}R_{\mathfrak {p}}}$, i.e.

${\displaystyle \mu _{\mathfrak {p}}(M)=\operatorname {dim} _{k({\mathfrak {p}})}(M_{\mathfrak {p}}/{\mathfrak {p}}M).}$

• Rao, R.A.; Ischebeck, F. (2005). Ideals and Reality: Projective Modules and Number of Generators of Ideals. Deutschland: Physica-Verlag.
• Swan, Richard G. (1967). "The number of generators of a module". Math. Mathematische Zeitschrift. 102 (4): 318–322. doi:10.1007/BF01110912.