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Divisor :: isRegular

isRegular -- Checks to see if R mod the given ideal is regular

Synopsis

Description

Returns true if R/I is regular where R is the ambient ring of I, otherwise it sets to false.
i1 : R = QQ[x, y, z]

o1 = R

o1 : PolynomialRing
i2 : I = ideal(x * y - z^2 )

                  2
o2 = ideal(x*y - z )

o2 : Ideal of R
i3 : isRegular( I )

o3 = false
i4 : R = QQ[x, y, u, v]

o4 = R

o4 : PolynomialRing
i5 : I = ideal(x * y - u * v)

o5 = ideal(x*y - u*v)

o5 : Ideal of R
i6 : isRegular( I )

o6 = false
i7 : R = QQ[x, y, z]

o7 = R

o7 : PolynomialRing
i8 : J = ideal( x )

o8 = ideal x

o8 : Ideal of R
i9 : isRegular( J )

o9 = true
If IsGraded is set to true (default false) then it treats I as an ideal on Proj R. In particular, singularities at the origin (corresponding to the irrelevant ideal) are ignored.
i10 : R = QQ[x, y, z]

o10 = R

o10 : PolynomialRing
i11 : I = ideal(x * y - z^2 )

                   2
o11 = ideal(x*y - z )

o11 : Ideal of R
i12 : isRegular(I, IsGraded => true)

o12 = true
i13 : R = QQ[x, y, u, v]

o13 = R

o13 : PolynomialRing
i14 : I = ideal(x * y - u * v)

o14 = ideal(x*y - u*v)

o14 : Ideal of R
i15 : isRegular(I, IsGraded => true)

o15 = true

Ways to use isRegular :