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# acyclicClosure -- Compute the acyclic closure of a DGAlgebra.

- Usage:
    `B = acyclicClosure(A)`
- Inputs:
  - A, an instance of the type DGAlgebra,
- 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:
  - B, an instance of the type DGAlgebra, The acyclic closure of the DG Algebra A up to homological degree provided in the EndDegree option (default value is 3).

## Description

    i1 : R = ZZ/101[a,b,c]/ideal{a^3,b^3,c^3}

    o1 = R

    o1 : QuotientRing

    i2 : A = koszulComplexDGA(R);

    i3 : B = acyclicClosure(A,EndDegree=>3)

    o3 = {Ring => R                                  }
          Underlying algebra => R[T ..T ]
                                   1   6
                                     2     2     2
          Differential => {a, b, c, a T , b T , c T }
                                       1     2     3

    o3 : DGAlgebra

    i4 : toComplex(B,8)

          1      3      6      10      15      21      28      36      45
    o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R   <-- R

         0      1      2      3       4       5       6       7       8

    o4 : Complex

    i5 : B.diff

                                               2     2     2
    o5 = map (R[T ..T ], R[T ..T ], {a, b, c, a T , b T , c T , a, b, c})
                 1   6      1   6                1     2     3

    o5 : RingMap R[T ..T ] <-- R[T ..T ]
                    1   6         1   6

## See also

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

## Ways to use acyclicClosure:

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

## For the programmer

The object acyclicClosure is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/DGAlgebras.m2:2035:0.