Integral $p$-adic non-abelian Hodge theory for small representations

Let $\text{frakX}$ be a smooth $p$-adic formal scheme over ${\text{calO}}_C$ with rigid generic fiber $X$. In this paper, we construct a new period sheaf $\text{calO}{\widehat{\text{bC}}}_{\text{pd}}^{+}$ on $X_{\text{proet}}$ and use it to establish an integral $p$-adic Simspon correspondence for small $\text{OXp}$-representations on $X_{\text{proet}}$ and small Higgs bundles on ${\text{frakX}}_{\text{et}}$ which is compatible with the works on rational level. In particular, for a small $\text{OXp}$-representations $\text{calL}$ with induced Higgs bundle $(\text{calH},{\theta }_{\text{calH}})$, we provide a canonical morphism $\text{HIG}(\text{calH},{\theta }_{\text{calH}})\to \text{rR}{\nu }_{\ast }\text{calL}$ with a uniformly bounded $p^{\mathrm{\infty }}$-torsion cofiber. Finally, we shall use this canonical map to study an analogue of Deligne–Illusie decomposition with coefficients in small $\text{OXp}$-representations.