Robin Hartshorne's Algebraic Geometry Solutions

2
Applying the Ex. II-8.6- (a) again to the pair of liftings α i and µ i on U i , we have sections
ξ i of Der k (O X (U i ), I(U i )) for each U i with α i − µ = ξ i , that can also be seen as a section of
F ⊗ T on U i . Then, on U ij we have
β ij − β ′ ij = (α i − α j ) − (µ i − µ j ) = ξ i − ξ j ,
thus the cocycles {β ij } and {β ij ′ } give the same cohomology class in ˘H 1 (U, F ⊗ T ). This
last group is isomorphic to H 1 (X, F ⊗ T ) by (4.5). The converse is easy. This finishes the
proof.
□
4.11.