EndDegree => ..., default value 3, Option to specify the degree to stop computing the acyclic closure
StartDegree => ..., default value 1, Option to specify the degree to start computing the acyclic closure.
Outputs:
A, an instance of the type DGAlgebra, The acyclic closure of the ring R up to homological degree provided in the EndDegree option (default value is 3).
Description
This package always chooses the Koszul complex on a generating set for the maximal ideal as a starting point, and then computes from there, using the function acyclicClosure(DGAlgebra).
i1 : R = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^4-d^3}
o1 = R
o1 : QuotientRing
i2 : A = acyclicClosure(R,EndDegree=>3)
o2 = {Ring => R }
Underlying algebra => R[T ..T ]
1 7
2 2 3 2
Differential => {a, b, c, d, a T , b T , c T - d T }
1 2 3 4
o2 : DGAlgebra
i3 : A.diff
2 2 3 2
o3 = map (R[T ..T ], R[T ..T ], {a, b, c, d, a T , b T , c T - d T , a, b, c, d})
1 7 1 7 1 2 3 4
o3 : RingMap R[T ..T ] <-- R[T ..T ]
1 7 1 7
Ways to use this method:
acyclicClosure(Ring) -- Compute the acyclic closure of the residue field of a ring up to a certain degree