allgens -- List the generators of a given degree

Synopsis

Usage:

G = allgens A

Gd = allgens(A,d)
Inputs:
- A, an instance of the type DGAlgebra,
- d, an integer,
Outputs:
- G, a list, of all generators of A
- Gd, a list, of all generators of homological degree d.

Description

The DGAlgebra is constructed as a polynomial ring over a ground ring that is already a polynomial ring, and allgens includes the generators of the subring.

In invocations of bracket and bracketMatrix it is useful to refer to the generators by reference to the lists formed by allGens, rather than by trying to name them directly, since there is a confusion between generators of A.natural and generators of (flattenRing A.natural)_0.

i1 : kk = ZZ/101

o1 = kk

o1 : QuotientRing

i2 : S = kk[x,y,z]

o2 = S

o2 : PolynomialRing

i3 : R = S/ideal(x^2,y^2,z^2-x*y,x*z, y*z)

o3 = R

o3 : QuotientRing

i4 : lastCyclesDegree = 1

o4 = 1

i5 : KR = koszulComplexDGA(ideal R)

o5 = {Ring => S                                     }
      Underlying algebra => S[T ..T ]
                               1   5
                        2   2           2
      Differential => {x , y , - x*y + z , x*z, y*z}

o5 : DGAlgebra

i6 : A = acyclicClosure(KR, EndDegree => lastCyclesDegree);

This causes the generators of the acyclic closure to be computed up to homological degree lastCycleDegree+1, to kill the cycles in lastCycleDegree. If S is graded, then the generators of A.natural have degreeLength 2, with the first component the homological degree.

Since A.natural is a polynomial ring over S, gens A.natural only lists the generators of homological degree >=1, whereas allgens (made with fl

i7 : g = gens(A.natural)

o7 = {T , T , T , T , T , T , T , T , T , T  }
       1   2   3   4   5   6   7   8   9   10

o7 : List

i8 : g/degree

o8 = {{1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, {2, 3}, {2, 3},
     ------------------------------------------------------------------------
     {2, 3}}

o8 : List

i9 : allgens A

o9 = {x, y, z, T , T , T , T , T , T , T , T , T , T  }
                1   2   3   4   5   6   7   8   9   10

o9 : List

i10 : G3 = allgens(A,2)

o10 = {T , T , T , T , T  }
        6   7   8   9   10

o10 : List

i11 : G3/degree

o11 = {{2, 3}, {2, 3}, {2, 3}, {2, 3}, {2, 3}}

o11 : List

Ways to use allgens :

allgens(DGAlgebra)
allgens(DGAlgebra,ZZ)

For the programmer

The object allgens is a method function.