acyclicClosure(Ring) -- Compute the acyclic closure of the residue field of a ring up to a certain degree

Function: acyclicClosure
Usage:

A = acyclicClosure(R)
Inputs:
- R, a ring,
Optional inputs:
- 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

The source of this document is in DGAlgebras.m2:2057:0.