If M is a graded module over a ring S, then the S2-ification of M is ∑d ∈ZZ H0((sheaf M)(d)), which may be computed as limd->∞ Hom(S/Id,M), where Id is any sequence of ideals contained in higher and higher powers of S+. There is a natural restriction map f: M = Hom(S,M) →Hom(Id,M). We compute all this using the ideals Id generated by the d-th powers of the variables in S.
i1 : kk=ZZ/101 o1 = kk o1 : QuotientRing |
i2 : S = kk[a,b,c,d] o2 = S o2 : PolynomialRing |
i3 : M = truncate(3,S^1) o3 = image | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 | 1 o3 : S-module, submodule of S |
i4 : betti S2(0,M) 0 1 o4 = total: 1 20 0: 1 . 1: . . 2: . 20 o4 : BettiTally |
i5 : betti S2(1,M) 0 1 o5 = total: 1 20 0: 1 . 1: . . 2: . 20 o5 : BettiTally |
i6 : M = S^1/intersect(ideal"a,b,c", ideal"b,c,d",ideal"c,d,a",ideal"d,a,b") o6 = cokernel | cd bd ad bc ac ab | 1 o6 : S-module, quotient of S |
i7 : prune source S2(0,M) o7 = cokernel | cd bd ad bc ac ab | 1 o7 : S-module, quotient of S |
i8 : prune target S2(0,M) o8 = cokernel {-1} | d c b 0 0 0 0 0 0 0 0 0 | {-1} | 0 0 0 d c a 0 0 0 0 0 0 | {-1} | 0 0 0 0 0 0 d b a 0 0 0 | {-1} | 0 0 0 0 0 0 0 0 0 c b a | 4 o8 : S-module, quotient of S |