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MatFac15 :: MCMapproximation

MCMapproximation -- compute the MCM approximation

Synopsis

Description

The MCM approximation M=M’⊕F of the module N is upto twist a sufficientls high even syzygy module M’ of N plus possibly a free summand F together with an epimorphism φ: M -> N. The kernel P in the short exact sequence

0 →P →M →N →0

has finite projective dimension. Current implementation works only in case no free summand is needed.
i1 : kk=ZZ/10007;S=kk[x_0..x_4]

o2 = S

o2 : PolynomialRing
i3 : N=constructEx1(S);E=annihilator N;

o4 : Ideal of S
i5 : X=ideal(gens E*random(source gens E,S^{ -3}));SX=S/X;

o5 : Ideal of S
i7 : NX=N**SX;
i8 : phi=MCMapproximation NX;

o8 : Matrix
i9 : M=source phi;
i10 : P=ker phi;
i11 : betti res NX

             0  1  2  3  4  5  6
o11 = total: 6 13 18 18 18 18 18
          0: 6 13  3  .  .  .  .
          1: .  . 15 18  3  .  .
          2: .  .  .  . 15 18  3
          3: .  .  .  .  .  . 15

o11 : BettiTally
i12 : betti res M

              0  1  2  3  4  5  6
o12 = total: 18 18 18 18 18 18 18
         -1:  3  .  .  .  .  .  .
          0: 15 18  3  .  .  .  .
          1:  .  . 15 18  3  .  .
          2:  .  .  .  . 15 18  3
          3:  .  .  .  .  .  . 15

o12 : BettiTally
i13 : betti res P

              0  1
o13 = total: 25 18
         -1:  3  .
          0:  9 18
          1: 13  .

o13 : BettiTally

Ways to use MCMapproximation :