Syzygies

We now formalize the notion of cancellations of leading terms of polynomials, and

give an important example of how modules arise in algebraic geometry.

According to the Webster Third International Unabridged Dictionary, a syzygy is

the nearly straight-line configuration of three celestial bodies (as the sun, moon, and

earth during a solar or lunar eclipse) in a gravitational system.

Just as the sun or moon is obscured during an eclipse, leading terms of polynomials

are obscured by syzygies. The original Greek term συζυγ ´ια refers to a yoke,

conjunction, or copulation.

Definition 2.31 Let f1, . . . , fr ∈ k[x1, . . . , xn]. A syzygy among the f j is a relation

h1 f1 + h2 f2 + ·· ·+hr fr = 0

where (h1, . . . , hr ) ∈ k[x1, . . . , xn]r . The set of all such relations is denoted

Syz( f1, . . . , fr ) ⊂ k[x1, . . . , xn]r .

It is easy to check the following property of syzygies:

Proposition 2.32 Syz( f1, . . . , fr ) is a k[x1, . . . , xn]-submodule of k[x1, . . . , xn]r .

Corollary 2.35 (Generalized Hilbert Basis Theorem) The module of syzygies among

a set of polynomials is finitely generated.

Theorem 2.36 Let R be Noetherian and M ⊂ Rn an R submodule. Then M is
finitely generated.

 

Lemma2.37 Let M1 ⊂ M be R-modules such that M1 and M/M1 are both finitely
generated. Then M is also finitely generated.

Lemma 2.38 Suppose there exists a sequence of R-submodules

0 = M0 ⊂ M1 ⊂ M2 . . . ⊂ Mn = M

such that each Mi /Mi−1 is finitely generated. Then M is finitely generated.