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# approximationSequence -- Short exact sequence of the MCM approximation

- Usage:
    `E = approximationSequence M`
- Inputs:
  - M, a module,
- Outputs:
  - E, a complex,

## Description

The approximation sequence of a module M over a Gorenstein ring is the versal short exact sequence $$0\to P \to M' \to M \to 0$$ 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 : approximationSequence M

    o4 = M <-- cokernel | b2 a2 | <-- subquotient (| b2 a2 |, | b2 a2 |)

         1     2                      3

    o4 : Complex

## See also

- coApproximationSequence -- Short exact sequence of the MCM coapproximation

## For the programmer

The object approximationSequence is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MCMApproximations.m2:594:0.