During the post-COVID inflation episode, headline inflation increased rapidly and declined quickly while core inflation rose slowly and remained elevated. In several economies, most notably the U.S., disinflation proceeded without a significant rise in unemployment. We examine these patterns through a multi-sector setting with production network linkages. In our framework, an upstream sector produces intermediate inputs for a downstream sector, and the two sectors differ in the frequency of price adjustment. A negative productivity shock to the upstream sector requires relative prices to adjust, but heterogeneous nominal rigidities cause this adjustment to occur at different speeds across sectors. This generates persistent aggregate inflation even when monetary policy holds output at potential. The resulting aggregate Phillips curve features a relative-price gap term that plays the same analytical role as the exogenous cost-push shock in the one-sector model but arises endogenously from the interaction of sectoral shocks with the economy’s structure and responds to monetary policy. We revisit insights from previous work discussing how stabilizing the output gap rather than strictly targeting inflation comes close to optimal policy. We also discuss how omitting relative prices from empirical Phillips curve estimation produces misleading inferences about its slope, potentially leading to opposite and incorrect prescriptions for the policy response. We conclude by discussing how structural changes such as the energy transition, deglobalization, and technological progress could make relative-price disturbances more prominent, underscoring the importance of multi-sector frameworks for research and policy.