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

Chapter 3: Growth of the Betti sequence of the canonical module 39
3.1 The canonical module and Growth of its Betti
sequence
Let (R, m, k) be a Cohen-Macaulay local ring with unique maximal ideal m and
residue field k and of dimension d. A canonical module of R is a finitely generated
R-module ωR for which
Ext i R(k, ωR) ∼ ⎧
⎪⎨ 0 if i = d,
=
⎪⎩ k if i = d.
A canonical module of R, if it exists, is unique up to isomorphism. Obviously a
canonical module of R is maximal Cohen-Macaulay (MCM for short). If R is a
Gorenstein local ring, ωR = R, and if d = 0, ωR ∼ = E(k). It is known that if ωR
is a canonical module of R, then for any p ∈ Spec R, ωRp is a canonical module of
Rp. Also, it is easy to check that ωR, the m−adic completion of ωR, is a canonical
module of R and ωR/a ωR is a canonical module of R/a if a is an ideal which is
generated by an R-sequence. If the sequence is maximal, then dim R/a = 0 and so
ωR/a ωR ∼ = ER/a(k). We refer the reader to Chapter 3 of [16] for standard facts about
canonical modules.
Recall that by polynomial growth of a sequence {bi} we mean that there is an
integer d and a positive constant c such that bi ≤ cd i . Also a sequence {bi} has
exponential growth if there exist real numbers 1 < α < β such that α i < bi < β i for
all i ≫ 0.
Our investigation for studying the growth of the Betti sequence of ωR is started
with the following result: