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Macaulay2Doc :: degreesMonoid

degreesMonoid -- get the monoid of degrees

Description

Synopsis

  • Usage:
    degreesMonoid x
  • Inputs:
    • x, a list or an integera list of integers, or a single integer
  • Outputs:
    • the monoid with inverses whose variables have degrees given by the elements of x, and whose weights in the first component of the monomial ordering are minus the degrees. If x is an integer, then the number of variables is x, the degrees are all {}, and the weights are all -1.

This is the monoid whose elements correspond to degrees of rings with heft vector x, or, in case x is an integer, of rings with degree rank x and no heft vector; see heft vectors. Hilbert series and polynomials of modules over such rings are elements of its monoid ring over ZZ; see hilbertPolynomial and hilbertSeries The monomial ordering is chosen so that the Hilbert series, which has an infinite number of terms, is bounded above by the weight.

i1 : degreesMonoid {1,2,5}

o1 = monoid[T ..T , Degrees => {1..2, 5}, MonomialOrder => {MonomialSize => 32     }, DegreeRank => 1, Inverses => true, Global => false]
             0   2                                         {Weights => {-1, -2, -5}}
                                                           {GroupLex => 3          }
                                                           {Position => Up         }

o1 : GeneralOrderedMonoid
i2 : degreesMonoid 3

o2 = monoid[T ..T , Degrees => {3:{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
             0   2                                      {Weights => {3:-1} }
                                                        {GroupLex => 3     }
                                                        {Position => Up    }

o2 : GeneralOrderedMonoid

Synopsis

i3 : R = QQ[x,y,Degrees => {{1,-2},{2,-1}}];
i4 : heft R

o4 = {1, 0}

o4 : List
i5 : degreesMonoid R

o5 = monoid[T ..T , Degrees => {1, 0}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
             0   1                                      {Weights => {-1..0}}
                                                        {GroupLex => 2     }
                                                        {Position => Up    }

o5 : GeneralOrderedMonoid
i6 : S = QQ[x,y,Degrees => {-2,1}];
i7 : heft S
i8 : degreesMonoid S^3

o8 = monoid[T, Degrees => {{}}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 0, Inverses => true, Global => false]
                                                 {Weights => {-1}   }
                                                 {GroupLex => 1     }
                                                 {Position => Up    }

o8 : GeneralOrderedMonoid

See also

Ways to use degreesMonoid :