What does it represent by $\mathbb{Z}_p[1]$ in p-adic Hodge-Tate theory?
Is it called Tate object or Motives etc something like that?
I know it makes some geometric realization of cyclotomic character.
A free $R$-module $V$ of finite rank is a Galois representation if $V$ is equipped with cont. $R$-linear action $G_K \times V \to V$, where $G_K$ is Galois group. Then for cyclotomic character $\epsilon_p \leftrightarrow \mathbb{Z}_p(1) $ i.e., $\epsilon_p^i \leftrightarrow \mathbb{Z}_p(i)$. Then it is defined $$V(i):=V \otimes_{\mathbb{Z}_p} \mathbb{Z}_p(i), \ i \in \mathbb{Z}.$$
Can you explain please what does $\mathbb{Z}_p[1]$ or $\mathbb{Z}_p[i]$ represent in p-adic Hodge-Tate theory?