If F or G is a ring element of degree 0, it represents a trivial bundle of that rank.
i1 : X = PP^1
o1 = X
o1 : a flag bundle with subquotient ranks {2:1}
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i2 : OO_X(1) * OO_X(2)
o2 = a sheaf
o2 : an abstract sheaf of rank 1 on X
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i3 : chi oo
o3 = 4
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The sheaves can be on different varieties if one of the varieties is over the other.