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

For the programmer

The object approximationSequence is a function closure.

The source of this document is in MCMApproximations.m2:594:0.

approximationSequence -- Short exact sequence of the MCM approximation

Description

See also

For the programmer