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

Synopsis

Usage:
F+G
F++G
Operator: +
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 = PP^1

o1 = X

o1 : a flag bundle with ranks {1, 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 = projectiveSpace'(3,X,VariableName=>k)

o7 = Y

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

i8 : ch OO_Y(2)

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

           QQ[][h, H   ]
                    2,1
        ------------------[H   , H   , H   , k]
        (h + H   , h*H   )  1,1   1,2   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   , H   , k]
         (h + H   , h*H   )  1,1   1,2   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