An isochoric process is a thermodynamic process in which the volume of the closed system undergoing such a process remains constant ($\Delta V = 0$). The process in which pressure remains constant is called an isobaric process ($\Delta P = 0$). Therefore, the statement 'In an isochoric process, the pressure remains constant' is incorrect. The other statements are correct: in an adiabatic process there is no heat transfer (insulated), in an isothermal process temperature is constant, and an adiabatic process follows the relation $PV^\gamma = \text{constant}$ for an ideal gas.