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

Synopsis

• Scripted functor: Ext
• Usage:
Ext^i(M,N(>=d))
• Inputs:
• Outputs:
• , The R-module $\oplus_{j \geq d} Ext^i_X(M,N(j))$

Description

If M 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 = Proj R.

As an example, we consider the rational quartic curve in $P^3$.

 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(>=-3)) o6 = cokernel {-3} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 | {-3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a | 8 o6 : R-module, quotient of R i7 : Ext^0(IX,OO_X(>=-10)) o7 = cokernel {-1} | c 0 0 b 0 a 0 0 -d 0 0 0 0 0 0 0 | {-1} | -d c a 0 0 0 b 0 0 0 0 0 0 0 0 0 | {-1} | 0 -d -b 0 c 0 0 a 0 0 0 -d 0 0 0 0 | {-1} | 0 0 0 -d -d -c -c -b 0 0 0 0 0 -d 0 0 | {-1} | 0 0 0 0 0 0 0 0 c 0 0 b 0 a 0 0 | {-1} | 0 0 0 0 0 0 0 0 -2d c a 0 0 0 b 0 | {-1} | 0 0 0 0 0 0 0 0 0 -d -b 0 c 0 0 a | {-1} | 0 0 0 0 0 0 0 0 0 0 0 -2d -d -2c -c -b | 8 o7 : R-module, quotient of R

The method used may be found in: Smith, G., Computing global extension modules, J. Symbolic Comp (2000) 29, 729-746

If the vector space $Ext^i(M,N)$ is desired, see Ext^ZZ(CoherentSheaf,CoherentSheaf).