How far should central banks intervene?

To help stabilize financial markets and, above all, to support economic activity during the last crises, central banks have proven themselves reactive and creative, adopting so-called unconventional monetary policies (negative interest rates, massive purchases of public and private securities, etc.). How effective would their interventions have been without coordination with fiscal policy? These crises have reminded us of the complementarities between monetary policy and fiscal policy. But what is the limit to central bank purchases of public debt? What are the resulting risks? Is the independence of central banks threatened? Should we fear a return of inflation? Has the increase in the price of financial and real estate assets contributed to widening inequalities and does it not create a risk of financial instability? What more can central banks do? How can the climate constraint be added to their strategy?