next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
CompleteIntersectionResolutions :: complexity

complexity -- complexity of a module over a complete intersection

Synopsis

Description

The minimal resolution of a module over a complete intersection has betti numbers that grow as a polynomial of degree at most equal to the codimension-1. The complexity is one more than the degree of this polynomial.
i1 : setRandomSeed 0

o1 = 0
i2 : S = ZZ/101[a,b,c,d];
i3 : ff1 = matrix"a3,b3,c3,d3";

             1       4
o3 : Matrix S  <--- S
i4 : ff =ff1*random(source ff1, source ff1);

             1       4
o4 : Matrix S  <--- S
i5 : R = S/ideal ff;
i6 : M = highSyzygy (R^1/ideal"a2b2");
i7 : complexity M

o7 = 2
i8 : mf = matrixFactorization (ff, M)

o8 = {{7} | a   37b 0     0  |, {8} | 39a2 -29b -18a2 0    0    |}
      {6} | -b2 a2  -16b2 0  |  {8} | 39b2 39a  0     0    0    |
      {7} | 0   0   a     -b |  {8} | 0    0    39a2  -29b 0    |
                                {8} | 0    0    -29b2 -29a 41b2 |

o8 : List
i9 : complexity mf

o9 = 2
i10 : betti res (R^1/ideal"a2b2", LengthLimit=>10)

             0 1 2 3 4 5 6 7 8 9 10
o10 = total: 1 1 2 3 4 5 6 7 8 9 10
          0: 1 . . . . . . . . .  .
          1: . . . . . . . . . .  .
          2: . . . . . . . . . .  .
          3: . 1 2 1 . . . . . .  .
          4: . . . 2 4 2 . . . .  .
          5: . . . . . 3 6 3 . .  .
          6: . . . . . . . 4 8 4  .
          7: . . . . . . . . . 5 10

o10 : BettiTally

See also

Ways to use complexity :