ALMOST four years ago the Soviet "nationalities problem" reared into public view with the outbreak of ethnic violence between Azerbaijan and Armenia. At issue was the Armenian enclave of Nagorno-Karabakh within Azerbaijan. The Armenians said their rights and culture were being repressed; the Azeris felt the territorial integrity of their republic was at stake.In the years since, some 800 people have died in fighting between the two sides, which has included vicious pogroms against Armenians living in the Azerbaijani capital of Baku. A half-million refugees have fled the violence. The Soviet central government, under Mikhail Gorbachev, groped for a way to respond, sent in troops to keep the combatants apart, but generally failed to ease tensions. Now with central authority virtually out of the picture, a new peacemaking effort is under way, led by Russian President Boris Yeltsin. The early prospects are hopeful, though the persistence of animosity between Armenians and Azeris works against optimism. Mr. Yeltsin, accompanied by Nursultan Nazarbayev, president of Kazakhstan, touched all bases in the region, including Nagorno-Karabakh itself. He got commitments from local leaders to attend talks, which began Monday. Armenia, whose people voted overwhelmingly Sunday for independence, has apparently buried the issue of territorial claims on Nagorno-Karabakh. President Levon Ter-Petrosyan disavowed any such intentions. The central concerns of cultural rights and a measure of self rule for the Armenians who live in the enclave seem to be well on their way toward resolution. The Caucasus could be a test of the ability of the newly freed republics to work together to lessen ethnic and political tensions. Just to the north of Armenia and Azerbaijan, in Georgia, friction between pro-democracy forces and the government of President Zviad Gamsakhurdia threatens civil war. Yeltsin has made a good start at mediating. His efforts deserve international backing. If the republics are to move toward political and economic betterment and take their place in the community of nations, they will have to develop peaceful means of resolving conflicts among and within them.