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

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

o6 : List

i7 : analyze sheaf M

                 3   2   2
o7 = {{1, -1}, {a , b , b }, {1}  | 0 0 0 1 0 |, | a3 0  0  |}
                             {-1} | 0 0 0 0 1 |  | 0  b2 0  |
                                                 | 0  0  b2 |

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.

Source code

The source code from which this documentation is derived is in the file AnalyzeSheafOnP1.m2.

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(CoherentSheaf)" -- see analyze -- Compute the decomposition of a sheaf on P1
- "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(CoherentSheaf)" -- see showSheafOnP1 -- Prints the analysis of a sheaf on P1
- "showSheafOnP1(Module)" -- see showSheafOnP1 -- Prints the analysis of a sheaf on P1

For the programmer

The object AnalyzeSheafOnP1 is a package.