Compute the regularity of K[B] from the decomposition of the homogeneous monomial algebra K[B].
We assume that B=<b1,...,br> is homogeneous and minimally generated by b1,...,br, that is, there is a group homomorphism φ: G(B) →ℤ such that φ(bi) = 1 for all i.
In the case of a monomial curve an ad hoc formula for the regularity of the components is used (if R or B is given).
Specifying R:
i1 : a=5 o1 = 5 |
i2 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o2 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o2 : List |
i3 : R=QQ[x_0..x_3,Degrees=>B] o3 = R o3 : PolynomialRing |
i4 : dc=decomposeMA R o4 = HashTable{| -1 | => {ideal 1, | 4 |} } | 1 | | 1 | 2 | -2 | => {ideal (x , x ), | 3 |} | 2 | 1 0 | 2 | | 1 | => {ideal 1, | 1 |} | -1 | | 4 | 2 | 2 | => {ideal (x , x ), | 2 |} | -2 | 1 0 | 3 | 0 => {ideal 1, 0} o4 : HashTable |
i5 : regularityMA(dc,B) 2 2 o5 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o5 : List |
Specifying the decomposition dc:
i6 : a=5 o6 = 5 |
i7 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o7 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o7 : List |
i8 : R=QQ[x_0..x_3,Degrees=>B] o8 = R o8 : PolynomialRing |
i9 : dc=decomposeMA R o9 = HashTable{| -1 | => {ideal 1, | 4 |} } | 1 | | 1 | 2 | -2 | => {ideal (x , x ), | 3 |} | 2 | 1 0 | 2 | | 1 | => {ideal 1, | 1 |} | -1 | | 4 | 2 | 2 | => {ideal (x , x ), | 2 |} | -2 | 1 0 | 3 | 0 => {ideal 1, 0} o9 : HashTable |
i10 : regularityMA(dc,B) 2 2 o10 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o10 : List |
Specifying B:
i11 : a=5 o11 = 5 |
i12 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o12 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o12 : List |
i13 : regularityMA B 2 2 o13 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o13 : List |
Compare to
i14 : I=ker map(QQ[s,t],QQ[x_0..x_3],matrix {{s^a,t^a,s*t^(a-1),s^(a-1)*t}}) 4 3 3 2 2 2 2 3 3 4 o14 = ideal (x x - x x , x - x x , x x - x x , x x - x x , x x - x ) 0 1 2 3 2 1 3 0 2 1 3 0 2 1 3 0 2 3 o14 : Ideal of QQ[x , x , x , x ] 0 1 2 3 |
i15 : -1+regularity I o15 = 3 |