I thought the PKK was mostly actually in Turkey, not the other Kurdish groups. Erdowan has been trying to reislamize Turkey after Attaturk had made it mostly a secular state for many decades. Erdowan has been getting electoral pushback and probably decided he needed a war to build up national support.
From what I understand the PKK is both Syrian and Turkish. It is a Maoist/Marxist organization and is on the State department’s list of terrorist groups. President Obama set up the US military alliance with them and the Syrians Kurds went along since there was a common enemy in ISIS. The group which the President negotiated a pause with Turkey are the Syrian Kurds who are democratic and do not want an independent state but rather an autonomous region within a newly reconstituted Syria. They had already started last year after ISIS’s defeat to negotiate with Assad and had even set up an office in Damascus. See: THE KURDS, ASSAD AND THE SHAPE OF A FUTURE SYRIA in the Jerusalem Post 10/17/2019