AnalyzeSheafOnP1 -- Describe a graded module over k[x,y] without 0-dimensional torsion

Description

Any sheaf on P1 is the direct sum of line bundles-- and cyclic skyscraper sheaves represented by modules of the form k[x,y]/(l^m) where l is an kirreducible homogeneous polynomial and m is a non-negative integer. The routine "analyze" computes the twists and the annihilators l^m that appear in the decomposition, starting from a coherent sheaf on P1 or a graded module over a polynomial ring on 2 variables.

i1 : k = ZZ/5

o1 = k

o1 : QuotientRing

i2 : S = k[a,b]

o2 = S

o2 : PolynomialRing

i3 : M = S^1/ideal(a^3)++S^{-1}/(ideal b^2)++S^1/(ideal b^2)++ S^{-1,1}

o3 = cokernel {0}  | a3 0  0  |
              {1}  | 0  b2 0  |
              {0}  | 0  0  b2 |
              {1}  | 0  0  0  |
              {-1} | 0  0  0  |

                            5
o3 : S-module, quotient of S

i4 : L = analyze M;

i5 : twists = L_0

o5 = {1, -1}

o5 : List

i6 : anns = L_1

          2     3 2
o6 = {1, b , -2a b }

o6 : List

i7 : analyze sheaf M

                    2   3 2
o7 = {{1, -1}, {1, b , a b }, {1}  | 0 0 0 1 0 |, | 1 0  0    |}
                              {-1} | 0 0 0 0 1 |  | 0 b2 0    |
                                                  | 0 0  a3b2 |

o7 : List

Caveat

The script uses a linear nonzerodivisor, which would not exist over a finite field in the case where every point of P1 is the support of one of the skyscraper components.

Author

David Eisenbud <[email protected]>

Version

This documentation describes version 0.1 of AnalyzeSheafOnP1.

Citation

If you have used this package in your research, please cite it as follows:

@misc{AnalyzeSheafOnP1Source,
  title = {{AnalyzeSheafOnP1: decompose a Sheaf on P1. Version~0.1}},
  author = {David Eisenbud},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- analyze -- Compute the decomposition of a sheaf on P1
- doubleDualMap -- map from a module to its double dual
- isNZD -- tests whether a ring element is a non zerodivisor on a module
- killH0 -- removes 0-dimensional torsion
- showSheafOnP1 -- Prints the analysis of a sheaf on P1
Methods
- analyze(Module) -- see analyze -- Compute the decomposition of a sheaf on P1
- doubleDualMap(Module) -- see doubleDualMap -- map from a module to its double dual
- isNZD(RingElement,Module) -- see isNZD -- tests whether a ring element is a non zerodivisor on a module
- killH0(Module) -- see killH0 -- removes 0-dimensional torsion
- showSheafOnP1(Module) -- see showSheafOnP1 -- Prints the analysis of a sheaf on P1

For the programmer

The object AnalyzeSheafOnP1 is a package, defined in AnalyzeSheafOnP1.m2.

The source of this document is in AnalyzeSheafOnP1.m2:125:0.