Will a shift in the average total cost curve mean that the marginal cost curve will also shift up?

I understand the basic concept of marginal cost and average total cost. I am currently learning about costs curve for an oligopoly

I read online that the marginal cost is not determined by fixed costs, as the derivative of them is zero.

Based on this, is it possible for the average total cost curve to shift upwards without the marginal cost curve also shifting upwards? For example, when the fixed costs increase (rent goes up for example), would this cause a shift in the average cost curve upwards? And would marginal cost not increase?

Consider a shift from the total cost function $$C_1(q)=q$$ to $$C_2(q)=1+q$$.
Then the average cost functions shifts upward from $$AC_1(q)=1$$ to $$AC_2(q)=1/q+1$$.
But the marginal cost function which was $$MC_1(q)=1$$ remains unchanged at $$MC_2(q)=1$$.