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

## Synopsis

• Usage:
B = acyclicClosure(A)
• Inputs:
• Optional inputs:
• EndDegree => ..., default value -1, 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 : ChainComplex 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