Computes the set of associated primes for the ideal I.
i1 : R = ZZ/101[a..d]; |
i2 : I = intersect(ideal(a^2,b),ideal(a,b,c^5),ideal(b^4,c^4))
4 4 2 4
o2 = ideal (b , b*c , a c )
o2 : Ideal of R
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i3 : associatedPrimes I
o3 = {ideal (b, a), ideal (c, b)}
o3 : List
|
i4 : R = ZZ/7[x,y,z]/(x^2,x*y); |
i5 : I=ideal(0_R); o5 : Ideal of R |
i6 : associatedPrimes I
o6 = {ideal(x), ideal (y, x)}
o6 : List
|
In general, the associated primes are found using Ext modules: The associated primes of codimension i of I and Ext^i(R^1/I,R) are identical, as shown in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235.
primaryDecomposition also computes the associated primes. After doing a primaryDecomposition, calling associatedPrimes requires no new computation, and the list of associated primes is in the same order as the list of primary components returned by primaryDecomposition.
If the ideal is a monomial ideal, then a more efficient method is used. This monomial ideal code was written by Greg Smith and Serkan Hosten. The above comments about primary decomposition hold in this case too.
i7 : R = QQ[a..f]; |
i8 : I = monomialIdeal ideal"abc,bcd,af3,a2cd,bd3d,adf,f5"
2 4 3 5
o8 = monomialIdeal (a*b*c, a c*d, b*c*d, b*d , a*d*f, a*f , f )
o8 : MonomialIdeal of R
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i9 : ass I
o9 = {monomialIdeal (a, b, f), monomialIdeal (a, d, f), monomialIdeal (b, c,
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f), monomialIdeal (b, d, f), monomialIdeal (c, d, f), monomialIdeal (a,
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c, d, f)}
o9 : List
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i10 : primaryDecomposition I
2 5 5
o10 = {monomialIdeal (a , b, a*f, f ), monomialIdeal (a, d, f ),
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3
monomialIdeal (b, c, f), monomialIdeal (b, d, f ), monomialIdeal (c,
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4 3 4 5
d , d*f, f ), monomialIdeal (a, c, d , f )}
o10 : List
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Original author: C. Yackel, http://faculty.mercer.edu/yackel_ca/.