HH DGAlgebra -- The homology algebra of a DG algebra

Function: homology
Usage:

HA = homology A

HA = HH A
Inputs:
- A, an instance of the type DGAlgebra,
Outputs:
- HA, a ring, the homology algebra HH_*(A), as a graded-commutative ring

Description

Returns the homology algebra of A, caching the result inside A so subsequent calls are fast. This is a convenience wrapper around homologyAlgebra that supports the HH shorthand.

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 : HA = HH A
Finding easy relations           :  -- used 0.0408756s (cpu); 0.0357256s (thread); 0s (gc)

o3 = HA

o3 : QuotientRing

i4 : numgens HA

o4 = 10

i5 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))

o5 = {1, 4, 6, 3}

o5 : List

For DG algebras with even-degree generators (such as acyclic closures of non-regular rings), call homologyAlgebra directly and supply GenDegreeLimit and RelDegreeLimit explicitly.

Ways to use this method:

HH DGAlgebra -- The homology algebra of a DG algebra

The source of this document is in DGAlgebras/doc.m2:8166:0.

HH DGAlgebra -- The homology algebra of a DG algebra

Description

See also

Ways to use this method: