A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for
L
L
a line bundle on a smooth toric variety
X
X
over a field of positive characteristic, the direct image
F
∗
L
F_*L
under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators
Diff
(
1
)
(
L
,
L
)
≅
Hom
O
X
(
1
)
(
F
∗
L
,
F
∗
L
)
\operatorname {Diff}^{(1)} (L,L)\cong \operatorname {Hom}_{O_{X^{(1)}}}(F_*L,F_*L)
, and
T
T
-linearized sheaves.