It seems to me that there are two ways of thinking about how monetary policy would react to fiscal stimulus. One approach would be to ask: "What is the optimal Fed response to fiscal stimulus?" And the answer to that question is rather obvious; the Fed should act in such a way as to completely neutralize the impact of fiscal stimulus, i.e. make sure the multiplier is precisely zero. This is because the Fed has some optimal level of expected AD growth in mind, and that level should not change just because fiscal policy changed. So if the Fed is doing its job, which means if it is always targeting expected AD growth at what it sees as the optimal rate, then it will try to completely offset fiscal stimulus and the expected fiscal multiplier will be precisely zero. That's why fiscal stimulus almost disappeared from graduate textbooks in recent years.
IMO, this is the most important paragraph in Scott Sumner's recent post on the multiplier. I had thought I had known what the Keynesian multiplier was but now I realize that I was considering only one case: the increase in nominal GDP due to an increase in nominal aggregate demand caused by an increase in government spending. But read his post to see how much actual confusion he uncovers in the debate.