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

## Synopsis

• Function: acyclicClosure
• Usage:
A = acyclicClosure(R)
• 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:
• 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