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

## Synopsis

• Usage:
Ext^i(M,N)
• Scripted functor: Ext
• Inputs:
• Outputs:
• , The global Ext module ExtiX(M,N)

## 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 bd ac b2 | {3} | -b -a 0 0 | {3} | d c -b -a | {3} | 0 0 -d -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).