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

Let $X$ be a smooth p-adic formal scheme over $O_C$ with rigid generic fiber X. In this paper, we construct a new integral period sheaf $O{\hat{C}}_{\mathrm{pd}}^{+}$ on $X_{\mathrm{pro}\acute{e}t}$ and use it to establish an integral p-adic Simpson correspondence for small ${\hat{O}}_X^{+}$-representations on $X_{\mathrm{pro}\acute{e}t}$ and small Higgs bundles on $X_{\acute{e}t}$, which recovers rational p-adic Simpson correspondence for small coefficients after inverting p (at least in the good reduction case). Moreover, for a small ${\hat{O}}_X^{+}$-representation $L$ with induced Higgs bundle $(H,{\theta }_H)$, we provide a natural morphism $\mathrm{HIG}(H,{\theta }_H)\to R{\nu }_{⁎}L$ with a bounded $p^{\infty }$-torsion cofiber. Finally, we shall use this natural map to study an analogue of Deligne–Illusie decomposition with coefficients in small ${\hat{O}}_X^{+}$-representations.