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cycles -- Chosen cycle representatives for homology generators of a DG algebra

Description

When acyclicClosure, killCycles, or homologyAlgebra adjoin new generators to kill homology of a DG algebra A, the chosen cycle representatives are cached on the output so the exact Tate construction can be inspected afterwards. The key is A.cycles on a DG algebra, or HA.cache.cycles on the output of homologyAlgebra.

i1 : R = ZZ/101[a, b, c, d] / ideal(a^3, b^4, c^5, d^6)

o1 = R

o1 : QuotientRing
 
i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra
 
i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))

o3 = {1, 4, 6, 4, 1}

o3 : List
 
i4 : HA = homologyAlgebra A
Finding easy relations           :  -- used 0.0210438s (cpu); 0.0185669s (thread); 0s (gc)

o4 = HA

o4 : PolynomialRing, 4 skew commutative variable(s)
 
i5 : numgens HA

o5 = 4
 
i6 : HA.cache.cycles

       2       3       4       5
o6 = {a T   , b T   , c T   , d T   }
         1,1     1,2     1,3     1,4

o6 : List

The entries of HA.cache.cycles are elements of A.natural: cycles chosen to represent the generators of HA as an algebra. These are the same representatives that would be used as the differentials of newly adjoined generators in acyclicClosure.

See also

For the programmer

The object cycles is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:1232:0.