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

Chapter 3: Growth of the Betti sequence of the canonical module 41
Proof. 1. For each i > 0, put Bi = Tor R i (ωR, k). Applying the functor ωR ⊗R −
to the given exact sequence, we obtain a long exact sequence part of which
is Tor R j+1(ωR, M) → B m j+1 → B n j → Tor R j (ωR, M). Note that Bi = k ri where
ri = β R i (ωR). Thus for each j indeed we have the following exact sequence,
Tor R j+1(ωR, M) → k mrj+1 → k nrj → Tor R j (ωR, M).
Since Tor R t (ωR, M) = Tor R t+1(ωR, M) = 0, it yields that nrt = mrt+1.
In particular, if Tor R j (ωR, M) = 0 for all j ≤ t, it turns out recursively that
rt = (n/m) t−1 r1, that is, β R t (ωR) = (n/m) t−1 · β R 1 (ωR).
2. This is proved the same way as (1), using ExtR(ωR, −).
3. Since we have Tor R j (ωR, M) = 0 for all j ≥ t + 1, similar to (1) we get that
β R j+t(ωR) = (n/m) j−1 · β R t+1(ωR) for all j ≥ 1. Now note that if n > m, then
we get the exponential growth of {β R i (ωR)} with the base n/m. When n < m,
R will be Gorenstein because then β R i (ωR) = 0 for i ≫ 0. Furthermore, in the
case n = m we would obtain constant Betti numbers. This would give the
polynomial growth, assuming of course that each β R i (ωR) = 0.
4. The same interpretations as that of (3).
In the following some examples are provided.
Example 3.1.2. Let (R, m, k) be a local CM ring which has a canonical module ωR.
Let M be an R-module such that Tor R
i (ωR, M) = 0, ∀i ≫ 0 (e.g., proj.dimM < ∞).
1. If the sequence 0 → k 2 → M → k 3 → 0 is exact, then {β R i (ωR)} has exponential
growth, provided of course that β i R (ωR) = 0 for i ≫ 0