Ring ^ List -- make a free module

• Operator: ^

• Usage:

  M = R^{i,j,k,...}

• Inputs:
  • R, a ring
  • {i,j,k, ...}, a list, of integers or lists of integers
• Outputs:
  • a module, , a free module over R whose generators have degrees -i, -j, -k, ...

i1 : R = QQ[a..d]

o1 = R

o1 : PolynomialRing

i2 : R^{-1}

      1
o2 = R

o2 : R-module, free, degrees {1}

i3 : R^{-1,2:-2,-3}

      4
o3 = R

o3 : R-module, free, degrees {1..2, 2..3}

i4 : R = QQ[x,y,z,Degrees=>{{1,0},{1,-1},{1,-2}}]

o4 = R

o4 : PolynomialRing

i5 : R^{{1,2}}

      1
o5 = R

o5 : R-module, free, degrees {{-1, -2}}