We investigate heating mechanisms of the neutral intergalactic medium (IGM) around galaxies at $z\sim 10$ with a focus on cosmic ray (CR) effects.
Besides conventional X-ray heating and direct CR heating via ionization and Coulomb collisions, we examine CR-driven resistive heating caused by return currents induced by streaming CRs.
We include both Coulomb and charge-neutral collisions in modeling the resistivity and account for their dependence on temperature and ionization degree.
Numerical results show that CR-driven resistive heating can dominate over the other mechanisms, rapidly increasing the IGM temperature to $10^4 \, \mathrm{K}$ near the galaxy and creating a hot region with $T\gtrsim 10^2 \, \mathrm{K}$ extending to $\sim 30 \, \mathrm{kpc}$.
The size and ionization state of this region depend on the star formation rate, suggesting that future $21$-$\mathrm{cm}$ line observations could probe the impact of CRs on the early IGM.

