Compute the regularity of K[B] from the decomposition of the homogeneous monomial algebra K[B].
We assume that B=<b_{1},...,b_{r}> is homogeneous and minimally generated by b_{1},...,b_{r}, that is, there is a group homomorphism \phi : G(B) \to \mathbb{Z} such that \phi(b_{i}) = 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 : regularityMA R
2 2
o4 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}}
1 0 | 2 | 1 0 | 3 |
o4 : List
|
Specifying a monomial algebra:
i5 : a=5 o5 = 5 |
i6 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}
o6 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}}
o6 : List
|
i7 : M=monomialAlgebra B
ZZ
o7 = ---[x ..x ]
101 0 3
o7 : MonomialAlgebra generated by {{5, 0}, {0, 5}, {1, 4}, {4, 1}}
|
i8 : regularityMA M
2 2
o8 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}}
1 0 | 2 | 1 0 | 3 |
o8 : List
|
Specifying the decomposition dc:
i9 : a=5 o9 = 5 |
i10 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}
o10 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}}
o10 : List
|
i11 : R=QQ[x_0..x_3,Degrees=>B] o11 = R o11 : PolynomialRing |
i12 : dc=decomposeMonomialAlgebra R
o12 = HashTable{| -1 | => {ideal 1, | 4 |} }
| 1 | | 1 |
2
| -2 | => {ideal (x , x ), | 3 |}
| 2 | 1 0 | 2 |
0 => {ideal 1, 0}
| 1 | => {ideal 1, | 1 |}
| -1 | | 4 |
2
| 2 | => {ideal (x , x ), | 2 |}
| -2 | 1 0 | 3 |
o12 : HashTable
|
i13 : regularityMA(B,Decomposition=>dc)
2 2
o13 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}}
1 0 | 2 | 1 0 | 3 |
o13 : List
|
Specifying B:
i14 : a=5 o14 = 5 |
i15 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}
o15 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}}
o15 : List
|
i16 : regularityMA B
2 2
o16 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}}
1 0 | 2 | 1 0 | 3 |
o16 : List
|
Compare to
i17 : 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
o17 = 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
o17 : Ideal of QQ[x ..x ]
0 3
|
i18 : -1+regularity I o18 = 3 |
The object regularityMA is a method function with options.