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

Appendix A
Algorithms for our criterion for
linear resolutions
In this appendix we develop some algorithms which support the theoretical aspects
of Chapter 2. The programming in CoCoA (or any other Computer Algebra System)
is then straightforward.
Let K be a field, I = (f1, . . . , fm) be a graded ideal of S = K[x1, · · · , xr] generated
in a single degree, say d. Consider the Rees algebra of I
R(I) =
j≥0
I j t j = S[f1t, . . . , fmt] ⊆ S[t].
Let T = S[t1, · · · , tm]. Consider the natural surjective homomorphism of bigraded
K-algebras ϕ : T −→ R(I) with ϕ(xi) = xi for i = 1, . . . , r and ϕ(tj) = fjt for
j = 1, . . . , m. Furthermore, let deg(xi) = (0, 1) and deg(tj) = (1, 0). So we can easily
write
• R(I) = T/P,
• R(I)(k,j) = (I k )kd+j
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