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

Chapter 2: Linear resolution of powers of an ideal 36
x > t t > x
DegRevLex | in(P2) |= 34; (1, 2) : 6, (2, 2) : 3 | in(P2) |= 33; (1, 2) : 6, (2, 2) : 3
Lex | in(P2) |= 32; (1, 2) : 6, (2, 2) : 3 | in(P2) |= 32; (1, 2) : 6, (2, 2) : 3
Table 2.3: Count of elements of in(P2) with degx > 1 for the ideal of (2.21).
J2 has 8 generators so let T = Q[x1, · · · , x6, t1, · · · , t8] and in DegRevLex order. Let
P2 be the defining ideal of the Rees ring of J2, i.e., R(J2) = T/P2. One can observe
that P2 is consisting of 11 elements of bidegree (1, 1), 1 element of bidegree (2, 2),
2 elements of bidegree (3, 0), and 2 elements of bidegree (4, 0). In Table 2.3 we give
a report of elements of in(P2) w.r.t. term ordering x > t or t > x in either Lex or
DegRevLex order. Due to existence of guys with x-degree > 1 we are unable to apply
Theorem 2.1.2 (in four ordinary term orderings discussed in Table 2.3 at least) to
deduce the linear resolution of powers of J2. Hence we try to make a use of Algorithm
7 in order to find a suitable upper triangular bi-change of x and t that fulfils the
requirements of our criterion. It was interesting (for us at least) to report that after
more or less 122, 000 times of tests we were unable to find such a bi-change. Indeed
we believe that J2 is one whose powers have non-linear resolution. In fact, reg(J2) = 3
and we have checked that
reg(J 2 2 ) = 7, reg(J 3 2 ) = 10, reg(J 4 2 ) = 13, reg(J 5 2 ) = 16 and reg(J 6 2 ) = 19
it attracts our interests to the following question:
Question 2.4.8. Is it true that reg(J k 2 ) = 3k + 1, ∀k ≥ 2?