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# isLocallyFree -- checks whether a toric reflexive sheaf is locally free

- Usage:
    `isLF = isLocallyFree E`
- Inputs:
  - E, a vector bundle on a toric variety (Klyachko's description)
- Outputs:
  - isLF, a Boolean value, true if the reflexive sheaf is locally free, false otherwise
- Consequences:
  - The result of isLocallyFree will be stored as a cacheValue (missing documentation) in E.cache#isLF.

## Description

Contrary to what the name suggests, ToricVectorBundle may well encode a toric reflexive sheaf that is not locally free. isLocallyFree permits to check whether a toric reflexive sheaf is locally free, that is, locally a direct sum of invertible sheaves, that is, a vector bundle.
isLocallyFree calls internally the methods isLocallyWeil and toricChernCharacter.

    i1 : A3 = fan coneFromVData matrix {{1,0,0},{0,1,0},{0,0,1}};

    i2 : filtMat = apply( { {{1,0},{0,1}}, {{0,1},{1,0}}, {{1,1},{1,0}} }, matrix);

    i3 : filtStep = apply( { {{0,1}}, {{0,1}}, {{0,1}} }, matrix);

    i4 : E = toricVectorBundle (2,A3,filtMat,filtStep);

    i5 : details E

    o5 = HashTable{| 0 | => (| 1 0 |, | 0 1 |)}
                   | 0 |     | 0 1 |
                   | 1 |
                   | 0 | => (| 0 1 |, | 0 1 |)
                   | 1 |     | 1 0 |
                   | 0 |
                   | 1 | => (| 1 1 |, | 0 1 |)
                   | 0 |     | 1 0 |
                   | 0 |

    o5 : HashTable

    i6 : isLocallyFree E

    o6 = false

    i7 : P112 = ccRefinement transpose matrix {{1,0},{0,1},{-1,-2}};

    i8 : D = {1,0,0};

    i9 : L = toricVectorBundle(1, P112, toList(3: matrix{{1_QQ}}), apply( D, i-> matrix{{-i}}));

    i10 : details L

    o10 = HashTable{| -1 | => (| 1 |, | -1 |)}
                    | -2 |
                    | 0 | => (| 1 |, 0)
                    | 1 |
                    | 1 | => (| 1 |, 0)
                    | 0 |

    o10 : HashTable

    i11 : isLocallyWeil L

    o11 = true

    i12 : toricChernCharacter L

    o12 = HashTable{| -1 0 | => {| 1 |}  }
                    | -2 1 |     | 0 |
                    | 1 -1 | => {| 0   |}
                    | 0 -2 |     | 1/2 |
                    | 1 0 | => {0}
                    | 0 1 |

    o12 : HashTable

    i13 : isLocallyFree L

    o13 = false

## Caveat

This method works for any toric reflexive sheaf on a toric variety, whose fan is covered by cones of maximal dimension.

## See also

- ToricVectorBundleKlyachko -- the class of all toric vector bundles in Klyachko's description
- toricChernCharacter -- computes the toric Chern character of a toric vector bundle
- isLocallyWeil -- checks whether a toric reflexive sheaf is locally Weil

## Ways to use isLocallyFree:

- `isLocallyFree(ToricVectorBundleKlyachko)`

## For the programmer

The object isLocallyFree is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PositivityToricBundles.m2:1023:0.