I’ve had it explained to me that the Fed’s “purchasing” has only ever been requirements that Federal Reserve Banks must purchase legitimate US bonds and mortgage securities for resale. In essence, there’s been no expansion of the monetary supply equal in any way to the amount of these pass through purchases that ultimately are purchased away from those Banks if not kept.
The concern with the above is with the mortgage securities, in which part of that pass through purchasing becomes inflationary, in my understanding.
It sure looks like the money base has expanded pretty rapidly to me. research.stlouisfed.org