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=decomposeMonomialAlgebra 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=decomposeMonomialAlgebra 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 |