At first, raising interest rates in the face of high deficit-driven inflation can slow inflation down, which makes it seem like it’s working. This is because the Fed can potentially reduce the rate of bank lending, and thus slow the economy down a bit, even as those fiscal deficits keep pouring in. In other words, they’re not affecting the primary cause of inflation, but they’re subduing enough other things that they’re able to push back against the primary cause, indirectly.
Over time, however, raising interest rates and keeping them high in an environment where runaway government deficits and high government debts are causing inflation runs the risk of exacerbating inflation. High interest rates on large amounts of government debt (>100% of GDP) will result in even bigger runaway deficits, because now they are dealing with ballooning interest payments on the debt, and this ironically pushes more money into the economy.
However, high interest rates also exacerbate deficit-driven inflation, specifically during eras with unusually large sovereign debts and deficits (e.g. >100% debt-to-GDP and >7% structural deficits-to-GDP). Each increase in interest rates puts some disinflationary pressure on the private sector, but also results in even larger public sector deficits pouring money into the economy. If those public sector deficits are big enough, then high interest rates can actually be inflationary.
One example with insightful graphs: https://www.lynalden.com/inflation-vs-interest-rates/
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