The
K
K
-theory of smooth schemes is
A
1
\mathbf {A}^{1}
-invariant. We show that this remains true over finite fields if one replaces the affine line by the Frobenius line, i.e., the non-commutative algebra where multiplication with the variable behaves like the Frobenius. Emerton had shown that over regular rings the Frobenius line is left coherent. As a technical ingredient for our theorem, but also of independent interest, we extend this and show that merely assuming finite type (or just
F
F
-finite), the Frobenius line is right coherent.