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