Lemma of Nakayama

The Nakayama lemma , named after the Japanese mathematician Tadashi Nakayama , is the following theorem of commutative algebra :

It is a finitely generated non-trivial - module and an ideal that in the Jacobson radical of lies. Then is .${\ displaystyle M}$${\ displaystyle R}$${\ displaystyle {\ mathfrak {a}}}$${\ displaystyle R}$${\ displaystyle {\ mathfrak {a}} M \ neq M}$

proof

We accept . Let it be a minimal generating system of . Since is nontrivial, follows and . ${\ displaystyle {\ mathfrak {a}} M = M}$${\ displaystyle \ {u_ {1}, \ ldots, u_ {n} \}}$${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle n \ geq 1}$${\ displaystyle u_ {n} \ not = 0}$

According to the assumption , there would then be an equation of the form with , therefore . ${\ displaystyle u_ {n} \ in {\ mathfrak {a}} M}$${\ displaystyle u_ {n} = \ sum _ {i = 1} ^ {n} a_ {i} u_ {i}}$${\ displaystyle a_ {i} \ in {\ mathfrak {a}}}$${\ displaystyle (1-a_ {n}) u_ {n} = \ sum _ {i = 1} ^ {n-1} a_ {i} u_ {i}}$

Since it is in the Jacobson radical, the factor is a unit. The generating system is therefore not minimal and thus the assumption is refuted. ${\ displaystyle a_ {n}}$${\ displaystyle 1-a_ {n}}$

Inferences

• If a finitely generated module, a sub-module and an ideal, then applies${\ displaystyle M}$${\ displaystyle R}$${\ displaystyle N}$${\ displaystyle {\ mathfrak {a}} \ subset J (R)}$
${\ displaystyle M = {\ mathfrak {a}} M + N \ \ Rightarrow \ M = N}$.

This conclusion, which is equivalent to the above lemma and is therefore also referred to as the lemma of Nakayama, can be used to raise bases:

• Let it be a local ring , its maximal ideal and the remainder class field .${\ displaystyle R}$${\ displaystyle {\ mathfrak {m}}}$${\ displaystyle \ kappa: = R / {\ mathfrak {m}}}$
Then are archetypes of a base - vector space , then generate the modulus .${\ displaystyle x_ {1}, \ ldots, x_ {n}}$${\ displaystyle \ kappa}$ ${\ displaystyle M / {\ mathfrak {m}} M}$${\ displaystyle x_ {i}}$${\ displaystyle M}$

Individual evidence

1. Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), sentence 2.5.24
2. Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , Lemma IV.2.2