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# AbstractSheaf + AbstractSheaf -- (direct) sum of two abstract sheaves

- Operator: +
- Usage:
    `F+G`
    `F++G`
- Inputs:
  - F, an abstract sheaf,
  - G, an abstract sheaf,
- Outputs:
  - an abstract sheaf, the (direct) sum of $F$ and $G$. If one of the arguments is an integer, it represents the trivial sheaf of that rank.

## Description

    i1 : X = abstractProjectiveSpace 1

    o1 = X

    o1 : a flag bundle with subquotient ranks {2:1}

    i2 : OO_X(1) + OO_X(2)

    o2 = a sheaf

    o2 : an abstract sheaf of rank 2 on X

    i3 : chi oo

    o3 = 5

    i4 : 1 + OO_X(1)

    o4 = a sheaf

    o4 : an abstract sheaf of rank 2 on X

    i5 : chi oo

    o5 = 3

The sheaves can be on different varieties if one of the varieties is over the other.

    i6 : ch OO_X(1)

    o6 = 1 + H
              2,1

             QQ[][h, H   ]
                      2,1
    o6 : ---------------------
         (- h - H   , -h*H   )
                 2,1      2,1

    i7 : Y = abstractProjectiveSpace'(3,X,VariableName=>k)

    o7 = Y

    o7 : a flag bundle with subquotient ranks {3, 1}

    i8 : ch OO_Y(2)

                    2   4 3
    o8 = 1 + 2k + 2k  + -k
                        3

                     QQ[][h, H   ]
                              2,1
                 ---------------------[H   ..H   , k]
                 (- h - H   , -h*H   )  1,1   1,3
                         2,1      2,1
    o8 : ----------------------------------------------------
         (- H    - k, - H    - H   k, - H    - H   k, -H   k)
             1,1         1,2    1,1      1,3    1,2     1,3

    i9 : OO_Y(2) ++ OO_X(1)

    o9 = a sheaf

    o9 : an abstract sheaf of rank 2 on Y

    i10 : ch oo

                              2   4 3
    o10 = 2 + (2k + H   ) + 2k  + -k
                     2,1          3

                      QQ[][h, H   ]
                               2,1
                  ---------------------[H   ..H   , k]
                  (- h - H   , -h*H   )  1,1   1,3
                          2,1      2,1
    o10 : ----------------------------------------------------
          (- H    - k, - H    - H   k, - H    - H   k, -H   k)
              1,1         1,2    1,1      1,3    1,2     1,3

    i11 : chi ooo

    o11 = 12

## Ways to use this method:

- AbstractSheaf + AbstractSheaf -- (direct) sum of two abstract sheaves
- `AbstractSheaf + QQ`
- `AbstractSheaf + RingElement`
- `AbstractSheaf + ZZ`
- `QQ + AbstractSheaf`
- `RingElement + AbstractSheaf`
- `ZZ + AbstractSheaf`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Schubert2.m2:1857:0.