isWellDefined(DGAlgebra) -- Check that a DG algebra has correct structure and a differential that squares to zero

Function: isWellDefined
Usage:

b = isWellDefined A
Inputs:
- A, an instance of the type DGAlgebra,
Outputs:
- b, a Boolean value, True when A has all expected structural keys with the correct types, the base ring of A.natural is compatible with A.ring, the differential lowers homological degree by one on every generator, and d^2 = 0 holds on every generator of A.natural.

Description

Checks three conditions in order. First, a structural validation: the DG algebra has keys ring, natural, diff, and Degrees with the right types, and the length of A.Degrees equals the number of generators of A.natural. Second, every generator g of A.natural has d(g) of homological degree |g| - 1. Third, d^2(g) = 0 on every generator — Leibniz then extends this to all of A.natural.

When debugLevel > 0 the routine prints a diagnostic line for each failing check.

i1 : R = QQ[x, y, z]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o2 : DGAlgebra

i3 : isWellDefined A

o3 = true

A DG algebra with A.diff == {} (no differential set) passes the check trivially — the second and third conditions apply only to generators with a stored differential.

Ways to use this method:

isWellDefined(DGAlgebra) -- Check that a DG algebra has correct structure and a differential that squares to zero

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4943:0.

isWellDefined(DGAlgebra) -- Check that a DG algebra has correct structure and a differential that squares to zero

Description

See also

Ways to use this method: