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

Description

If F or G is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.

Both F and G must be coherent sheaves on the same projective variety or scheme $X$.

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

i1 : S = QQ[a..d]

o1 = S

o1 : PolynomialRing
 
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 |
              {-2} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
              {-2} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
              {-2} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |

                            4
o6 : R-module, quotient of R
 
i7 : Ext^0(IX, OO_X(>=-10))

o7 = cokernel {-1} | b  0  a  0  d   -c 0  0   0  0   0  0  c2  0   0   |
              {-1} | 0  c  0  a  0   0  0  d   d  0   0  b  -d2 -bd b2  |
              {-1} | -d -d -c -b 0   0  -c 0   0  d   0  0  0   c2  -ac |
              {-1} | 0  0  0  0  c   0  0  b   0  a   0  0  0   0   0   |
              {-1} | 0  0  0  0  -2d c  a  0   0  0   b  0  0   0   0   |
              {-1} | 0  0  0  0  0   -d -b 0   c  0   0  a  0   0   0   |
              {-1} | 0  0  0  0  0   0  0  -2d -d -2c -c -b 0   0   0   |

                            7
o7 : R-module, quotient of R

The algorithm used may be found in:

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

See also

Ways to use this method:

The source of this document is in Varieties/doc-functors.m2:600:0.