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

Chapter 2: Linear resolution of powers of an ideal 25
I, in any of these ways, emphasizes the structure of the polynomial relations amongst
the elements of a generating set of I. Here we start with a matrix of presentation of
the ideal I = (f1, . . . , fm), and using elimination theory directly or via Gröbner basis
computations, one seeks to describe P.
As it was indicated in [67] studying the Rees algebra of an ideal I is focused on the
degrees of a generating set for the presentation ideal P and seeks to obtain those
equations from the syzygies of I. The ideal P, which we refer to as the equations of
R[It], is graded (in fact, a bigraded ideal of T ):
P = P1 + P2 + · · · ,
where P1 is the R-module of linear forms aiti such that aifi = 0. The module
Pr is the module of syzygies of the r-products of the fi.
Example 2.2.1. From [67, Example 1.2], if the ideal I is generated by a regular
sequence f1, · · · , fm, the equations of R(I)are nice:
⎛
R(I) ∼ ⎜
= S[t1, . . . , tm]/I2 ⎝ t1 · · · tm
f1 · · · fm
⎞
⎟
⎠ .
In other words, P is generated by the Koszul relations of the fi.
Now let R be a ring and M a finitely generated R-module. We discuss presenta-
tions of the symmetric algebra of M. The symmetric algebra SR(M), or simply S(M),
is the algebra
SR(M) = TR(M)/(x ⊗ y − y ⊗ x, x, y ∈ M)
where TR(M) is the tensor algebra of M over R. It is convenient to give it in terms
of generators and relations. This arises directly from a free presentation of M, as