# Commutative algebra

Yoshifumi Tsuchimoto

LEMMA 05.1   Let

be an exact sequence of -modules. Then we have

DEFINITION 05.2   Let be a graded algebra. We assume
1. (Length of as an module is finite.)
2. is generated by homogeneous elements where .
Then for any graded finite -module , We define its Hilbert series as

PROPOSITION 05.3   Under the assumption of the definition above, The Hilbert series is a rational function on . More precisely, we have

PROPOSITION 05.4   If a graded algebra is generated by of degree over a ring with , there exists a polynomial such that

We call the Hilbert polynomial of .

COROLLARY 05.5   Let be a Noetherian local ring. Let be an ideal of definition (That means, there exists such that holds.) We put . Then there exists a polynomial such that holds for .

DEFINITION 05.6   Under the hypothesis of the Corollary above, we define the Samuel function of as .

THEOREM 05.7 (Nakayama's lemma, or NAK)   Let be a commutative ring. Let be an -module. We assume that is finitely generated (as a module) over . That means, there exists a finite set of elements such that

holds. If an ideal of satisfies

that is,

then there exists an element such that

holds. If furthermore is contained in (the Jacobson radical of ), then we have .

PROOF.. Since , there exists elements such that

holds. In a matrix notation, this may be rewritten as

with , . Using the unit matrix one may also write :

Now let be the adjugate matrix of . In other words, it is a matrix which satisfies

Then we have

On the other hand, since modulo , we have for some . This clearly satisfies