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ResidualIntersections :: genericArtinNagata

genericArtinNagata -- Generic Artin nagata

Synopsis

Description

forms an ideal F generated by s generic linear combinations of the generators of I in the degree of the highest degree generator, and computes K = F:I. If the codimension of K is not equal to s, returns {-1,K}. Otherwise returns {codepth R/K,K}, where codepth R/K is the deviation from Cohen-Macaulayness. Thus genericArtinNagata(s,I)_0 = 0 means that K is an s-residual intersection of codim s and R/K is Cohen-Macaulay.

If I is a monomial idal, the function residualCodims I returns the list of codimensions s for which there might be a residual intersection of codimension s.

In the following example, all the generic residual intersectionsa are Cohen-Macaulay, until we get to the 6-residual intersection, which cannot be codim 6 because there are only 5 variables.

i1 : setRandomSeed 0

o1 = 0
i2 : S = ZZ/101[a,b,c,d,e]

o2 = S

o2 : PolynomialRing
i3 : I = minors(2, random(S^2, S^{3:-1}))

               2              2                      2                  
o3 = ideal (45a  + 24a*b - 50b  + 10a*c + 48b*c + 45c  - 49a*d - 50b*d -
     ------------------------------------------------------------------------
                2                                  2     2              2  
     10c*d + 23d  + 3a*e - 7b*e + 8c*e - 4d*e + 16e , 22a  + 45a*b + 17b  +
     ------------------------------------------------------------------------
                     2                             2                        
     36a*c + b*c + 6c  - 31a*d - 13b*d - 4c*d + 22d  - 27a*e - 30b*e + 44c*e
     ------------------------------------------------------------------------
                 2     2             2                      2                
     + 21d*e + 4e , 24a  + 2a*b + 35b  + 44a*c + 15b*c + 34c  - 41a*d + 18b*d
     ------------------------------------------------------------------------
                  2                                      2
     + 48c*d + 49d  + 41a*e - 15b*e + 16c*e - 13d*e - 32e )

o3 : Ideal of S
i4 : apply(5, i-> (genericArtinNagata(i+2,I))_0)

o4 = {0, 0, 0, 0, -1}

o4 : List
i5 : I = randomShellableIdeal(S,2,4)

o5 = monomialIdeal (b*c, b*d, c*d*e)

o5 : MonomialIdeal of S
i6 : residualCodims I

o6 = {3, 4, 5, 6}

o6 : List
i7 : apply(5, i-> (genericArtinNagata(i+2,I))_0)

o7 = {0, 0, 0, 0, -1}

o7 : List

Caveat

Ways to use genericArtinNagata :

For the programmer

The object genericArtinNagata is a method function.