Given a henselian pair
(
R
,
I
)
(R, I)
of commutative rings, we show that the relative
K
K
-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace
K
→
T
C
K \to \mathrm {TC}
. This yields a generalization of the classical Gabber–Gillet–Thomason–Suslin rigidity theorem (for mod
n
n
coefficients, with
n
n
invertible in
R
R
) and McCarthy’s theorem on relative
K
K
-theory (when
I
I
is nilpotent).
We deduce that the cyclotomic trace is an equivalence in large degrees between
p
p
-adic
K
K
-theory and topological cyclic homology for a large class of
p
p
-adic rings. In addition, we show that
K
K
-theory with finite coefficients satisfies continuity for complete noetherian rings which are
F
F
-finite modulo
p
p
. Our main new ingredient is a basic finiteness property of
T
C
\mathrm {TC}
with finite coefficients.