Betti numbers of modules over Noetherian rings with ... - IPM

Chapter 1: Preliminaries 2
ideal. If R is local, then we let m denote the maximal ideal and we let K denote
the residue class field R/m. If R is homogeneous, then m will denote the irrelevant
maximal ideal m = R1 ⊕ R2 ⊕ · · · .
When R is a homogeneous K-algebra, automatically I is a homogeneous ideal of R.
Furthermore M stands for a finitely generated R-module unless otherwise specified.
1.1 Betti numbers and graded Betti numbers
The Betti numbers of a finitely generated module M over a commutative Noethe-
rian local unit ring R are the minimal numbers bi for which there exists a long exact
sequence
bn fn
0 → R −→ R bn−1 b1 → . . . → R f1 b0
f0
−→ R −→ M → 0,
which is called a minimal free resolution of M. The Betti numbers are uniquely
determined by requiring that bi be the minimal number of generators of Kerfi−1 for
all i ≥ 0.
Graded modules arise naturally in homology. For example for every integer i,
there exists an ith singular homology group of a space Hi(X), and usually the “total
homology” of the space is considered to be the direct sum ⊕iHi(X). This makes the
“total” homology of X a Z- module graded over the natural numbers N. There are
several reasons to support studying Graded Betti numbers in detail. A minimal free
resolution of a finitely generated graded module M over a commutative Noetherian
N-graded ring R in which all maps are homogeneous module homomorphisms, i.e.,
the homomorphisms that map every homogeneous element to a homogeneous element