In this paper, we establish a duality for finite $t$-modules
and study its basic properties.
Our duality is the $\Bbb F_q[t]$-analogue of the Cartier duality,
where the multiplicative group $\Bbb G_m$ is replaced by the
Carlitz module $C$.
Finite $t$-modules are, roughly speaking, finite locally free
group schemes which are $\Bbb F_q[t]$-submodules of
abelian $t$-modules with scalar $t$-action on their tangent spaces.

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In fact, it is only for finite $v$-modules that we can define
the duality in a way with Dieudonn\'e theoretic flavor.

A typical case of our duality is supplied by division points of
Drinfeld modules and dual Drinfeld modules,
and is studied in some detail in Section 5.
In Section 6, some results on the duality of $\pi$-divisible groups are given.

One may hope to have such a duality for a wider class of $t$-modules,
namely, torsion points of abelian $t$-modules which do not have
scalar $t$-action on the tangent spaces,
such as higher Carlitz modules $C^{\otimes n}$.
But this would be possible only if the target $C$ of the pairing
was replaced by a tensor power $C^{\otimes n}$ with sufficiently large $n$.