getGenerators -- Cycles whose classes generate the homology algebra of a DG algebra

Usage:

cycleList = getGenerators A
Inputs:
- A, an instance of the type DGAlgebra,
Optional inputs:
- DegreeLimit => ..., default value -1, Last homological degree in which to search for algebra generators
- StartDegree => ..., default value 1, First homological degree in which to search for new algebra generators
- Verbosity => ..., default value 0, Option to control progress output from getGenerators
Outputs:
- cycleList, a list, a list of cycles in A.natural whose homology classes generate HH_*(A) as a graded-commutative algebra over A.ring

Description

Walks up the homological degrees of A, computing a basis of each homology module and selecting minimal new generators not already reachable by products of lower-degree cycles. The returned list is exactly the generating set used internally by homologyAlgebra to build its presentation.

This version of the function assumes all algebra generators of A lie in odd homological degree. When A has generators in even degree, a finite generating set need not exist, and the search depth must be bounded with EndDegree or GenDegreeLimit.

i1 : R = ZZ/101[a,b,c] / ideal(a^3, b^3, c^3, a^2*b^2*c^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {a, b, c}

o2 : DGAlgebra

i3 : netList getGenerators A

     +------------------+
     | 2                |
o3 = |a T               |
     |   1,1            |
     +------------------+
     | 2                |
     |b T               |
     |   1,2            |
     +------------------+
     | 2                |
     |c T               |
     |   1,3            |
     +------------------+
     |   2 2            |
     |a*b c T           |
     |       1,1        |
     +------------------+
     |   2 2            |
     |a*b c T   T       |
     |       1,1 1,2    |
     +------------------+
     | 2   2            |
     |a b*c T   T       |
     |       1,1 1,2    |
     +------------------+
     |   2 2            |
     |a*b c T   T       |
     |       1,1 1,3    |
     +------------------+
     |   2 2            |
     |a*b c T   T   T   |
     |       1,1 1,2 1,3|
     +------------------+
     | 2   2            |
     |a b*c T   T   T   |
     |       1,1 1,2 1,3|
     +------------------+
     | 2 2              |
     |a b c*T   T   T   |
     |       1,1 1,2 1,3|
     +------------------+

The generators above are the cycles used to present homologyAlgebra A; their classes span HH_i A in each positive degree.

i4 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))

o4 = {1, 4, 6, 3}

o4 : List

Ways to use getGenerators:

getGenerators(DGAlgebra)

For the programmer

The object getGenerators is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8433:0.

getGenerators -- Cycles whose classes generate the homology algebra of a DG algebra

Description

See also

Ways to use getGenerators:

For the programmer