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Macaulay2Doc :: Ext^ZZ(CoherentSheaf,SumOfTwists)

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

Synopsis

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 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(>=-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 Exti(M,N) is desired, see Ext^ZZ(CoherentSheaf,CoherentSheaf).

See also