  We study the space of $p$-compact operators $\mathcal K_p$, using the theory
of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated
to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$
(which is equal to $g_{p'}'$). This allows us to relate the theory of
$p$-summing operators with that of $p$-compact operators. With the results
known for the former class and appropriate hypothesis on $E$ and $F$ we prove
that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of
values of $p$ and $q$, and show that our results are sharp. We also exhibit
several structural properties of $\mathcal K_p$. For instance, we obtain that
$\mathcal K_p$ is regular, surjective, totally accessible and characterize its
maximal hull $\mathcal K_p^{max}$ as the dual ideal of the $p$-summing
operators, $\Pi_p^{dual}$. Furthermore, we prove that $\mathcal K_p$ coincides
isometrically with $\mathcal {QN}_p^{dual}$, the dual ideal of the quasi
$p$-nuclear operators.
