Let I⊂R be an ideal in a ring R, the image of a free R-module F. Let ReesI be the Rees algebra of I. Certain of the minimal primes of I are distinguished from the point of view of intersection theory: These are the ones that correspond to primes P⊂ReesI minimal among those containing I*ReesI---in other words, the isolated components of the support of the normal cone of I. The prime p corresponding to P is simply the kernel of the the induced map R →ReesI/P.
Each of these primes comes equipped with a multiplicity, which may be computed as the ratio degree(ReesI/P)/degree(R/p).
For these matters and their significance, see section 6.1 of the book “Intersection Theory,” by William Fulton, and the references there, along with the paper
“A geometric effective Nullstellensatz.” Invent. Math. 137 (1999), no. 2, 427--448 by Ein and Lazarsfeld.
i1 : T = ZZ/101[c,d]; |
i2 : D = 4; |
i3 : P = product(D, i -> random(1,T)) 4 3 2 2 3 4 o3 = - 50c - 2c d + 31c d - 7c*d - 36d o3 : T |
i4 : R = ZZ/101[a,b,c,d] o4 = R o4 : PolynomialRing |
i5 : I = ideal(a^2, a*b*(substitute(P,R)), b^2) 2 4 3 2 2 3 4 o5 = ideal (a , - 50a*b*c - 2a*b*c d + 31a*b*c d - 7a*b*c*d - 36a*b*d , ------------------------------------------------------------------------ 2 b ) o5 : Ideal of R |
i6 : ass I o6 = {ideal (b, a), ideal (c + 39d, b, a), ideal (c + 32d, b, a), ideal (c - ------------------------------------------------------------------------ 13d, b, a)} o6 : List |
i7 : primaryDecomposition I 2 2 2 2 2 2 o7 = {ideal (b , a*b, a ), ideal (c + 32d, b , a ), ideal (c - 13d, b , a ), ------------------------------------------------------------------------ 2 2 2 2 ideal (c - 23c*d + 6d , b , a )} o7 : List |
i8 : distinguished(I) o8 = {ideal (b, a)} o8 : List |
i9 : K = distinguishedAndMult(I) o9 = {{2, ideal (b, a)}} o9 : List |