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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.

i1 : S = QQ[a..d];
i2 : I = monomialCurveIdeal(S,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of S
i3 : R = S/I

o3 = R

o3 : QuotientRing
i4 : X = Proj R

o4 = X

o4 : ProjectiveVariety
i5 : IX = sheaf (module I ** R)

o5 = cokernel {2} | c2 ac bd b2 |
              {3} | -b 0  -a 0  |
              {3} | d  -b c  -a |
              {3} | 0  -d 0  -c |

                                         1           3
o5 : coherent sheaf on X, quotient of OO  (-2) ++ OO  (-3)
                                        X           X
i6 : Ext^1(IX,OO_X)

o6 = 0

o6 : QQ-module
i7 : Hom(IX,OO_X)

       16
o7 = QQ

o7 : QQ-module, free
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