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

Synopsis

Function: codim
Usage:

codim F
Inputs:
- F, a coherent sheaf, a coherent sheaf over a ProjectiveVariety X
Optional inputs:
- Generic (missing documentation) => ..., default value false,
Outputs:
- an integer

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))

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.

Ways to use this method: