getBasis -- Get a basis for a particular homological degree of a DG algebra.

Synopsis

Usage:

M = getBasis(n,A)
Inputs:
- n, an integer,
- A, an instance of the type DGAlgebra,
Optional inputs:
- Limit => ..., default value -1, Option to specify the maximum number of basis elements to return
Outputs:
- M, a matrix, The basis of the desired homological degree of the DG Algebra.

Description

This function is to allow for the retrieval of a basis of a particular homological degree of a DGAlgebra when the underlying algebra A.natural is multigraded. In the code, the homological grading is always the first integer in the degree tuple, and so this function returns a matrix consisting of all monomials in homological degree n.

i1 : R = ZZ/101[a..d, Degrees=>{1,1,1,2}]

o1 = R

o1 : PolynomialRing

i2 : A =  koszulComplexDGA(R)

o2 = {Ring => R                      }
      Underlying algebra => R[T ..T ]
                               1   4
      Differential => {a, b, c, d}

o2 : DGAlgebra

i3 : getBasis(3,A)

o3 = | T_1T_2T_3 T_1T_2T_4 T_1T_3T_4 T_2T_3T_4 |

                       1                4
o3 : Matrix (R[T ..T ])  <-- (R[T ..T ])
                1   4            1   4

Ways to use getBasis :

getBasis(ZZ,DGAlgebra)
getBasis(ZZ,Ring) -- Get a basis for a degree of a ring.

For the programmer

The object getBasis is a method function with options.