Consider an interacting field theory with Hamiltonian
$$H=H_0+V$$
where $H_0$ is the Hamiltonian of the free theory and $V$ is the added interaction. Now, I know the full Hamiltonian $H$ should be time-independent. Indeed, from the Heisenberg equation of motion we have
$$i\frac{\partial}{\partial t}H=[H,H]=0$$
However, since $H$ and $H_0$ do not generally commute I must have some time dependence in $H_0$.
$$i\frac{\partial}{\partial t}H_0=[H_0,H]\neq 0.$$
In for example the books by Weinberg and Peskin & Schroeder, they do implicitly assume time-independence of $H_0$, i.e that
$$\frac{\partial}{\partial t}H_0=0$$
when showing that the operator
$$U(t,t_0)=e^{iH_0(t-t_0)}e^{-iH(t-t_0)}$$
satisfies the Schrodinger Equation:
$$i\frac{\partial}{\partial t}U(t,t_0)=V_I(t)U(t,t_0), \quad V_I(t)=e^{iH_0(t-t_0)}Ve^{-iH_0(t-t_0)}$$
They write
$$\frac{\partial}{\partial t}U(t,t_0)=-ie^{iH_0(t-t_0)}(H-H_0)e^{-iH(t-t_0)}$$
which is what I would expect to get if I had $\frac{\partial}{\partial t}H_0=0$.
Can someone please tell me where I'm going wrong? I have a strong suspicion this boils down to me screwing up the differences between partial time derivatives and total time derivatives, so perhaps I should have
$$\frac{\partial}{\partial t}H_0=0, \quad \frac{d}{d t}H_0\neq 0$$
If this is the case I am then unsure whether the Heisenberg eom should involve a total or partial derivative (Peskin uses a partial).