# coApproximationSequence -- Short exact sequence of the MCM coapproximation

## Synopsis

• Usage:
E = coApproximationSequence M
• Inputs:
• M, ,
• Outputs:
• E, ,

## Description

The coapproximation sequence of a module M over a Gorenstein ring is the versal short exact sequence $$0\to M \to P \to M' \to 0$$ where M' is a maximal Cohen-Macaulay module and P is a module of finite projective dimension, as defined by Auslander and Buchweitz.

 i1 : S = ZZ/101[a,b]/ideal(a^3+b^3) o1 = S o1 : QuotientRing i2 : R = S/ideal(a*b) o2 = R o2 : QuotientRing i3 : M = R^1/(ideal vars R)^2 o3 = cokernel | a2 0 b2 | 1 o3 : R-module, quotient of R i4 : coApproximationSequence M 2 o4 = 0 <-- cokernel {-2} | b2 | <-- R <-- cokernel | a2 0 b2 | <-- 0 {-2} | a2 | 0 2 3 4 1 o4 : ChainComplex