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

Chapter 3: Growth of the Betti sequence of the canonical module 49
fact, ℓ(Tor R 1 (E, E)) = ℓ(E ⊗R E). Now the non-vanishing of E ⊗R E comes from the
surjection
E ⊗R E ↠ E/mE ⊗R E/mE
and the fact that the right hand side is an (d 2 =) 4-dimensional k-vector space.
3.4 Further analysis of vanishing of Ext and Tor
In this section we perform a homological analysis similar to Theorem 4.1.1, but
for different situations. In a sense, our situation is more general because we consider
sequences like (‡) and (‡†) where the ideals are possibly different from m. On the
other hand, we have to make different assumptions about the ideals, namely that J
in (‡) and the Jt in (‡†) are generated by regular sequences:
0 →
0 →
a
(R/I) → M →
s=1
a
(R/Is) → M →
s=1
b
(R/J) → 0 (‡†)
t=1
b
(R/Jt) → 0 (‡‡)
As it was mentioned by Hochster and Richert in [36, Remark 2.1] and as we
discovered we need to add some extra assumptions to situations (‡†) and (‡‡) which
is stated in the following:
t=1
1. J1, · · · , Jb are ideals generated by R-sequences.
2. Is ⊇ Jt for all s = 1, · · · , a and t = 1, · · · , b.
The reason to add such extra assumptions as it will be demonstrated in Lemma 3.4.1
is that when they hold the calculation of the relevant Ext and Tor modules are greatly
simplified, that is: