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Quasidegrees :: quasidegrees

quasidegrees -- compute the quasidegree set of a module

Synopsis

Description

The method quasidegrees takes a finitely generated module M over the polynomial ring that is presented by a monomial matrix and computes the quasidegree set of M. The quasidegrees of M are indexed by a list of pairs (v,F) where v is a vector and F is a list of vectors f1,...,fl. The pair (v,F) indexes the plane v+spanCF. The quasidegree set of M is the union of all such planes that the pairs (v,F) index.

If the input is an ideal I, then quasidegrees executes for the module R/I where R is the ring of I.

The following example computes the quasidegree set of ℚ[x,y]/<x2,y2> under the standard 2-grading.

i1 : A = matrix{{1,0},{0,1}}

o1 = | 1 0 |
     | 0 1 |

              2        2
o1 : Matrix ZZ  <--- ZZ
i2 : R = QQ[x,y, Degrees => entries transpose A]

o2 = R

o2 : PolynomialRing
i3 : I = ideal(x^2,y^2)

             2   2
o3 = ideal (x , y )

o3 : Ideal of R
i4 : M = R^1/I

o4 = cokernel | x2 y2 |

                            1
o4 : R-module, quotient of R
i5 : quasidegrees M

o5 = {{0, {}}, {| 0 |, {}}, {| 1 |, {}}, {| 1 |, {}}}
                | 1 |        | 0 |        | 1 |

o5 : List

The quasidegree set is given to be the points (0,1), (1,0), (1,1), and (0,0).

The next example takes R computes the quasidegrees of the above module after twisting R by multidegree (3,2).

i6 : R = R^{{-3,-2}}

               1
o6 = (QQ[x, y])

o6 : QQ[x, y]-module, free, degrees {{3, 2}}
i7 : M = R^1/I

o7 = cokernel {3, 2} | x2 y2 |

                                            1
o7 : QQ[x, y]-module, quotient of (QQ[x, y])
i8 : quasidegrees M

o8 = {{| 3 |, {}}, {| 3 |, {}}, {| 4 |, {}}, {| 4 |, {}}}
       | 3 |        | 2 |        | 2 |        | 3 |

o8 : List

The following demonstrates a quasidegree set that is not a finite number of points.

i9 : A = matrix{{1,0},{0,1}}

o9 = | 1 0 |
     | 0 1 |

              2        2
o9 : Matrix ZZ  <--- ZZ
i10 : R = QQ[x,y]

o10 = R

o10 : PolynomialRing
i11 : R = toGradedRing(A,R)

o11 = R

o11 : PolynomialRing
i12 : I = ideal(x^2*y,y^2)

              2    2
o12 = ideal (x y, y )

o12 : Ideal of R
i13 : M=R^1/I

o13 = cokernel | x2y y2 |

                             1
o13 : R-module, quotient of R
i14 : quasidegrees M

o14 = {{| 1 |, {}}, {| 0 |, {}}, {0, {| 1 |}}}
        | 1 |        | 1 |            | 0 |

o14 : List

In the above example, the quasidegree set of the module M consists of the points (1,1) and (0,1) along with the parameterized line (1,0)•t.

Ways to use quasidegrees :