If
M or
N is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.
M and
N must be coherent sheaves on the same projective variety or scheme
X.
As an example, we compute Hom_X(I_X,OO_X), and Ext^1_X(I_X,OO_X), for the rational quartic curve in
P3.
S = QQ[a..d]; |
I = monomialCurveIdeal(S,{1,3,4}) |
R = S/I |
X = Proj R |
IX = sheaf (module I ** R) |
Ext^1(IX,OO_X) |
Hom(IX,OO_X) |
The Ext^1 being zero says that the point corresponding to I on the Hilbert scheme is smooth (unobstructed), and vector space dimension of Hom tells us that the dimension of the component at the point I is 16.
The method used may be found in: Smith, G.,
Computing global extension modules, J. Symbolic Comp (2000) 29, 729-746
If the module
⊕d≥0 Exti(M,N(d)) is desired, see
Ext^ZZ(CoherentSheaf,SumOfTwists).