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

Chapter 3: Growth of the Betti sequence of the canonical module 52
2. Apply the covariant functor − ⊗R R/I to the given exact sequence and use
Lemma 3.4.1 (2) in the middle of the derived exact sequence to obtain the
following exact sequence for each i ≥ 0:
Tor R n
i+1(M, R/I) → (R/I)
b( i+1) R
→ (Tori (R/I, R/I)) a → Tor R i (M, R/I).
3. We have the following exact sequence
Ext i R(R/J, M) → (Ext i R(R/J, R/J)) b → (Ext i+1
R (R/J, R/I))a → Ext i+1
R
(R/J, M),
which is a part of the derived long exact sequence of the given exact sequence
after applying the covariant functor HomR(R/J, −). Now note that the vanish-
ing Ext i R(R/J, M) for i = t, t+1 and using the results provided by Lemma 3.4.1
we end up with
n
b
t
n
= a .
t + 1
4. This part is essentially the same as that of (3) with a slight change on the
direction of the derived exact sequence.
Finally we provide a proof for the general situation in (‡‡).
Theorem 3.4.3. Let R be a local ring and let M be a non zero R-module for which
a
b
0 → R/Is → M → R/Jt → 0 is an exact sequence where J1, · · · , Jb are ideals
s=1
t=1
generated by R-sequences of length n1, · · · , nb respectively. Then we have the follow-
ing:
1. Let Ext i R(M, R/Iu) = Ext i+1
R (M, R/Iu) = 0 for some 1 ≤ u ≤ a and for some
i, and let Iu ⊇ Jt for each t. Then
a
s=1
Ext i
R(R/Is, R/Iu) ∼ = (R/Iu) ( n1 i+1) · · · (R/Iu) ( nt i+1) .