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
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i2 : X = Proj R;
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i3 : T' = cotangentSheaf X
o3 = cokernel {2} | b a 0 0 0 0 -d2 c2 |
{2} | d 0 a 0 0 -c2 b2 0 |
{2} | -c 0 0 a 0 -d2 0 b2 |
{2} | 0 0 c d b2 a2 0 0 |
{2} | 0 d -b 0 c2 0 a2 0 |
{2} | 0 -c 0 -b d2 0 0 a2 |
6
o3 : coherent sheaf on X, quotient of OO (-2)
X
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i4 : numgens T'
o4 = 6
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i5 : module T'
o5 = cokernel {2} | b a 0 0 0 0 -d2 c2 |
{2} | d 0 a 0 0 -c2 b2 0 |
{2} | -c 0 0 a 0 -d2 0 b2 |
{2} | 0 0 c d b2 a2 0 0 |
{2} | 0 d -b 0 c2 0 a2 0 |
{2} | 0 -c 0 -b d2 0 0 a2 |
6
o5 : R-module, quotient of R
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