multigradedPolynomialRing -- produces polynomial rings with standard multigradings

Synopsis

Usage:

multigradedPolynomialRing L

multigradedPolynomialRing n
Inputs:
- L, a list, entries 1 less than the desired number of variables for each multidegree
- n, an integer, the rank of the grading, in the case with 2 variables for each multidegree
Optional inputs:
- CoefficientField => a ring, default value ZZ/32003, the coefficient field
- Variables => a string, default value "x", name to use for indexed variables
- Standard => a Boolean value, default value true
Outputs:
- a polynomial ring, a multigraded polynomial ring

Description

Given a list L, this function gives a $\ZZ^r$-graded polynomial ring (where $r$ is the length of L) containing L_i+1 variables of multidegree equal to the i-th basis vector of $\ZZ^r$, i.e. the coordinate ring of the product of projective spaces with dimensions the entries of L. Given an integer n it returns the coordinate ring of a product of n copies of $\PP^1$.

i1 : S = multigradedPolynomialRing({1,3,4})

o1 = S

o1 : PolynomialRing

i2 : gens S

o2 = {x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   }
       0,0   0,1   1,0   1,1   1,2   1,3   2,0   2,1   2,2   2,3   2,4

o2 : List

i3 : degrees S

o3 = {{1, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0,
     ------------------------------------------------------------------------
     0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}}

o3 : List

i4 : gens multigradedPolynomialRing 4

o4 = {x   , x   , x   , x   , x   , x   , x   , x   }
       0,0   0,1   1,0   1,1   2,0   2,1   3,0   3,1

o4 : List

By default the output will be a ring over ZZ/32003 in variables of the form x_(i,j). The coefficients can be changed using the option CoefficientField and the variable name with Variables (which takes a string). Setting the option Standard to false will produce variables with no indices, starting at a.

i5 : multigradedPolynomialRing({1,2},CoefficientField => ZZ/5,Variables=>"y")

     ZZ
o5 = --[y   ..y   , y   ..y   ]
      5  0,0   0,1   1,0   1,2

o5 : PolynomialRing

i6 : multigradedPolynomialRing(3,Standard=>false)

       ZZ
o6 = -----[a..f]
     32003

o6 : PolynomialRing

Caveat

The output of multigradedPolynomialRing is not compatible with some functions from the package TateOnProducts, such as cohomologyHashTable. Use productOfProjectiveSpaces instead.

Ways to use multigradedPolynomialRing :

multigradedPolynomialRing(List)
multigradedPolynomialRing(ZZ)

For the programmer