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Macaulay2Doc :: codim(CoherentSheaf)

codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety

Synopsis

Description

Computes the codimension of the support of F as given by dim(R) - dim(M) where M is the module representing F over the homogeneous coordinate ring R of X.
i1 : R = ZZ/31991[a,b,c,d];
i2 : I = monomialCurveIdeal(R,{1,3,5})

             2         2     2    3    2
o2 = ideal (c  - b*d, b c - a d, b  - a c)

o2 : Ideal of R
i3 : projplane = Proj(R)

o3 = projplane

o3 : ProjectiveVariety
i4 : II = sheaf module I

o4 = image | c2-bd b2c-a2d b3-a2c |

                                                         1
o4 : coherent sheaf on projplane, subsheaf of OO
                                                projplane
i5 : can = sheafExt^1(II,OO_projplane^1(-4))
-- ker (109) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (109) returned CacheFunction: -*a cache function*-
-- ker (109) called with Matrix: 0
--                             2
-- ker (109) returned Module: R
assert( ker(map(R^0,R^2,0)) === (R^2))
-- ker (110) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (110) returned CacheFunction: -*a cache function*-
-- ker (110) called with Matrix: 0
--                             3
-- ker (110) returned Module: R
assert( ker(map(R^0,R^{{-2}, {-1}, {-1}},0)) === (R^{{-2}, {-1}, {-1}}))

o5 = cokernel | c b a2 |
              | d c b2 |

                                                         2
o5 : coherent sheaf on projplane, quotient of OO
                                                projplane
i6 : codim can

o6 = 2

Caveat

The returned value is the usual codimension if R is an integral domain or, more generally, equidimensional.

See also