How should the Fisher relationship be interpreted?

The Fisher relationship says: Nominal interest rate = Real interest rate + Expected inflation In the long run, expected inflation is equal to inflation and we have: Nominal interest rate = Real interest rate + Inflation We can then have three interpretations of this relationship. 1. The usual interpretation: long-term inflation is determined by money supply growth; the real interest rate comes from the fundamental characteristics of the economy (potential growth, savings-investment equilibrium), and the nominal interest rate is endogenous, the sum of inflation and the real interest rate. 2. The neo -Fisherian interpretation: the central bank controls the nominal interest rate; the real interest rate comes from the fundamental characteristics of the economy and inflation is endogenous, equal to the gap between nominal and real interest rates. 3. The interpretation focused on monetary policy: the central bank sets the nominal interest rate according to inflation and the structural characteristics of the economy; inflation also results from structural characteristics of the economy; the real interest rate is endogenous, equal to the gap between nominal interest rates and inflation. What is the correct interpretation of the Fisher relationship? Since the 1990s, long-term inflation no longer varies with money supply growth, but with economic characteristics (e.g. labour market organisation); Especially since the implementation of quantitative easing, central banks control nominal interest rates, which depend, through central banks’ reaction, on inflation and structural characteristics of the economy (savings-investment equilibrium, therefore supply-demand relationship, functioning of the labour market, etc.); It therefore seems that the real interest rate is simply the gap between the nominal interest rate and inflation (interpretation focused on monetary policy); it depend s on the structural characteristics of the economy, not directly, but because monetary policy choices and inflation depend on them. For example, if there is an excess of savings, the central bank will set lower nominal interest rates, leading to lower real interest rates; but it also depends on central banks’ objectives.