SheafOfRings ^ List -- make a graded free coherent sheaf

Synopsis

• Operator: ^

• Usage:

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

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

Description

i1 : R = QQ[a..d]/(a*b*c*d)

o1 = R

o1 : QuotientRing

i2 : X = Proj R

o2 = X

o2 : ProjectiveVariety

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

        1           1           1
o3 = OO  (-1) ++ OO  (-2) ++ OO  (3)
       X           X           X

o3 : coherent sheaf on X, free

If i, j, ... are lists of integers, then they represent multi-degrees, as in graded and multigraded polynomial rings.

i4 : Y = Proj (QQ[x,y,z,Degrees=>{{1,0},{1,-1},{1,-2}}])

o4 = Y

o4 : ProjectiveVariety

i5 : OO_Y^{{1,2},{-1,3}}

        1              1
o5 = OO   (1, 2) ++ OO   (-1, 3)
       Y              Y

o5 : coherent sheaf on Y, free

i6 : degrees oo

o6 = {{-1, -2}, {1, -3}}

o6 : List

See also

• OO -- the structure sheaf
• Proj -- make a projective variety
• degrees -- degrees of generators
• graded and multigraded polynomial rings