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
|
The object genericArtinNagata is a method function.