The U.S.-Kurd alliance in Syria has a tangled history.

For U.S. forces in Syria, an on-again off-again alliance is very much on again.

The fighting around Sinaa prison in Hasaka, a city in northeastern Syria, has cast a spotlight on the predominantly Kurdish region, and also renewed questions of America’s role there.

The Syrian conflict dates to 2011, when a popular rebellion began against the government of President Bashar al-Assad, the country’s longtime dictator. The revolt started with peaceful demonstrations but quickly descended into a bloody conflict between rebels and government forces.

The Kurds, comprising about 10 percent of Syria’s population and concentrated in the northeast, largely stayed out of the fight.

But that changed in 2014, when jihadists of the Islamic State swept across eastern Syria and northern Iraq, creating a so-called caliphate the size of Britain. The rise of ISIS brought the United States directly into the conflict, with President Barack Obama assembling an international coalition to fight the group, and ordering airstrikes and dispatching the U.S. military to support local forces on the ground.

The coalition turned to a Kurdish militia that was already fighting the jihadists in Syria and formed a partnership that grew into the Syrian Democratic Forces, or S.D.F., and included fighters from other ethnic groups as well.

In March 2019, the S.D.F., backed by the United States, recaptured the last piece of ISIS-held territory. “We have won against ISIS,” President Donald J. Trump declared, adding “now it’s time for our troops to come back home.”

But the victory left a lot of unfinished business that set the stage for the events of the past week.

The S.D.F. fighters seized the opportunity to establish a wide measure of autonomy for themselves over northeastern Syria. They called their enclave Rojava and rapidly set up their own administration.

Diplomatically, the Kurdish-led administration has had only limited success, failing to win recognition from any country, including the United States. And the Kurdish-led push for political autonomy in Syria raised fears in Turkey, which sees the S.D.F. as deeply connected to the PKK, a Kurdish militant group considered a terrorist organization by Turkey and the United States that has fought a long, bloody insurgency against the Turkish state.

But Turkey declined to intervene, largely because of the thousands of American troops then working with the S.D.F., until October 2019, when President Trump abruptly ordered the withdrawal of most U.S. forces. That was seen as a green light for Turkey to invade, and it did, seizing control of a slice of northeastern Syria, which it still occupies.

More recently, the U.S. kept about 700 troops in northeastern Syria to help the S.D.F. battle the remnants of ISIS. But the withdrawal also provided the space that allowed the Islamic State to regroup, which helps explain why U.S. forces found themselves back in the fight in Syria this week.