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Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext

Synopsis

Description

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).

See also