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
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(>=-3))
o6 = cokernel {-3} | d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 a 0 0 0
{-3} | 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 a 0 0
{-3} | 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 a 0
{-3} | 0 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 a
{-3} | 0 0 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0
{-3} | 0 0 0 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0 0
{-3} | 0 0 0 0 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0 0
{-3} | 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 b 0 0 0 0
------------------------------------------------------------------------
0 0 0 0 |
0 0 0 0 |
0 0 0 0 |
0 0 0 0 |
a 0 0 0 |
0 a 0 0 |
0 0 a 0 |
0 0 0 a |
8
o6 : R-module, quotient of R
|
i7 : Ext^0(IX,OO_X(>=-10))
o7 = cokernel {-1} | c 0 b 0 -d 0 0 0 a 0 0 0 0 0 0 0 |
{-1} | -d c 0 0 0 0 0 0 0 b 0 0 a 0 0 0 |
{-1} | 0 -d 0 c 0 0 -d 0 0 0 0 0 -b a 0 0 |
{-1} | 0 0 0 0 c 0 b 0 0 0 a 0 0 0 0 0 |
{-1} | 0 0 -d -d 0 0 0 0 -c -c -d 0 0 -b 0 0 |
{-1} | 0 0 0 0 -2d c 0 0 0 0 0 b 0 0 a 0 |
{-1} | 0 0 0 0 0 -d 0 c 0 0 0 0 0 0 -b a |
{-1} | 0 0 0 0 0 0 -2d -d 0 0 -2c -c 0 0 0 -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
Exti(M,N) is desired, see
Ext^ZZ(CoherentSheaf,CoherentSheaf).