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MonomialAlgebras :: regularityMA

regularityMA -- Compute regularity from decomposition

Synopsis

Description

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

Ways to use regularityMA :

  • regularityMA(HashTable,List)
  • regularityMA(List)
  • regularityMA(PolynomialRing)