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Macaulay2Doc :: numgens(CoherentSheaf)

numgens(CoherentSheaf) -- the number of generators of the underlying module

Synopsis

Description

In Macaulay2, each coherent sheaf comes equipped with a module over the coordinate ring. In the homogeneous case, this is not necessarily the number of generators of the sum of twists H^0(F(d)), summed over all d, which in fact could be infinitely generated.
i1 : R = QQ[a..d]/(a^3+b^3+c^3+d^3)

o1 = R

o1 : QuotientRing
i2 : X = Proj R;
i3 : T' = cotangentSheaf X

o3 = cokernel {2} | c  0  0  d  0   a2  b2 0  |
              {2} | a  d  0  0  b2  -c2 0  0  |
              {2} | -b 0  d  0  a2  0   c2 0  |
              {2} | 0  b  a  0  -d2 0   0  c2 |
              {2} | 0  -c 0  a  0   -d2 0  b2 |
              {2} | 0  0  -c -b 0   0   d2 a2 |

                                         6
o3 : coherent sheaf on X, quotient of OO  (-2)
                                        X
i4 : numgens T'

o4 = 6
i5 : module T'

o5 = cokernel {2} | c  0  0  d  0   a2  b2 0  |
              {2} | a  d  0  0  b2  -c2 0  0  |
              {2} | -b 0  d  0  a2  0   c2 0  |
              {2} | 0  b  a  0  -d2 0   0  c2 |
              {2} | 0  -c 0  a  0   -d2 0  b2 |
              {2} | 0  0  -c -b 0   0   d2 a2 |

                            6
o5 : R-module, quotient of R

See also