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

## Synopsis

• Function: numgens
• Usage:
numgens F
• Inputs:
• F,
• Outputs:
• an integer, number of generators of the underlying module M of F

## 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