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CompleteIntersectionResolutions :: S2

S2 -- Universal map to a module satisfying Serre's condition S2

Synopsis

Description

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.

Since the result may not be finitely generated (this happens if and only if M has an associated prime of dimension 1), we compute only up to a specified degree bound b. For the result to be correct down to degree b, it is sufficient to compute Hom(I,M) where I ⊂(S+)r-b.
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

Caveat

Text S2-ification is related to computing cohomology and to computing integral closure; there are scripts in those packages that produce an S2-ification, but one takes a ring as argument and the other doesn’t produce the comparison map.

See also

Ways to use S2 :