President Obama had no real alternative to the air strikes he ordered last week against Islamic State (Isis) forces. The Yazidis and Christians of the region are in flight, and Kurdish territory proper, even the city of Erbil, could be at risk. Quite apart from the threat to the future of Iraq as a whole, the US and Britain have a humanitarian duty to the endangered minorities, and a debt of honour to the Kurds.

But that does not mean that this intervention is going to be either an easy or an effective one. For one thing, it comes after years in which the attempts of America and its allies to reshape Iraq have gone spectacularly wrong, illustrating again and again how disastrous has been the combination of ignorance and incompetence which has marked so many of the coalition’s efforts in that country.

For another, it comes at a time of extreme political volatility in every part of Iraq. There is of course no simple confrontation between Iraq and Isis. Instead there is a three-cornered contest between Iraq’s major regions, with further contests within each part. The most extreme of these regional contests, potentially, may be that between Isis and northern Sunni tribal and factional groups. But the Shia south is also divided, as Monday’s manoeuvres and counter-manoeuvres in Baghdad over the status of prime minister Nouri al-Maliki showed. So is Kurdistan, where longstanding rivalry between political parties and leading families has never been overcome, and where the peshmerga, the region’s army, although nominally united, still reflects those divisions.

The fact that Kurdistan has been a relative success story, compared with other parts of Iraq, has obscured the fact that it has many of the weaknesses of a typical oil-dependent state, including corruption, too many people on the government payroll, and a foreign labour force to do the menial as well as some of the expert jobs. This oil-based inflation of employment extends to the armed forces, criticised as too large, uncoordinated, too lightly equipped, and with poor logistical support. Professional armies do not run out of ammunition after a couple of days of combat, as some Kurd units are said to have done.

The US is gingerly re-inserting itself into the complex political and military landscape of Iraq, and Mr Obama is right to use very cautious language to describe his aims. His Republican critics are wrong to demand a bigger and tougher air campaign, which, as soon as it began to cause civilian casualties among ordinary Sunnis in the north, could have the counter-productive effect of strengthening Isis. The British MPs demanding RAF participation in the bombing as well as in the humanitarian missions, and a recall of parliament, are equally mistaken. Let’s not make ourselves out to be more important than we are. In any case, such bridges should be crossed when we come to them, if we do.

Much could go wrong. First, it is not guaranteed that American air power and the weapons it has said it will supply to the Kurds will redress the balance between Isis and the peshmerga. If they do, that will inevitably strengthen Kurdish demands for independence, or something close to it. Then, even if the Kurds are able to defend their own turf, it does not necessarily follow that they will have the capacity, or the will, to go beyond that and attack Isis in the Arab areas of the north.

Second, it is not guaranteed that the inclusive government the Americans, and probably the Iranians, want to see in Baghdad will come into being or be able to mount a serious challenge to Isis. Mr Maliki will remain an important political player, whether or not the post of prime minister goes to somebody else. A government plagued by Shia rivalries would not be well placed to reach out to the northern Sunni communities who are now either with Isis or just keeping their heads down.

America is right to intervene, but also right to see, this time, how difficult, and possibly indecisive, intervention will be and to look to Iraqis to come up, somehow, with a solution to their manifest difficulties. If they can.