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).