How is inflation determined in the long run?

In traditional theory, inflation is determined in the long run by money supply growth. But this theory no longer applies today, since the money supply is not controlled by central banks. Moreover, there is no longer any link between money supply growth and inflation. There are then two possibilities: Either the central bank sets the interest rate by a rule that makes it depend on the neutral real interest rate, inflation and the gap between inflation and target inflation; inflation in the long run is then the central bank's target inflation; Or the central bank has a nominal interest rate target (with yield curve control). We are then in a neo-Fisherism context , and it is the nominal interest rate that determines inflation in the long run .