Less than a week after the fourth bombing, a fifth bomb was uncovered in the basement of a building in another Russian city. It was disarmed before it could explode, and the bombers were arrested and identified. They turned out to be not Chechen terrorists but F.S.B. agents. Mr. Putin, himself a former head of the F.S.B., dismissed the notion that the bombings were a state-sponsored plot.

Yet suspicions that Moscow manipulates terrorism for its own purposes have re-emerged. In July, Novaya Gazeta, one of Russia’s few remaining independent newspapers, reported that the F.S.B. had been controlling the flow of jihadists from the North Caucasus to Syria, where many joined the Islamic State. The newspaper’s investigation found that the F.S.B. had established a “green corridor” allowing Islamist radicals to travel via Turkey, since Moscow would rather have these jihadists fighting in Syria than in Russia.

So much for leading the international effort against terrorism. Yet, that same month, President Obama said he was “encouraged” by a call from Mr. Putin to discuss Syria, and that this “offers us an opportunity to have a serious conversation.” Mr. Obama should not be fooled.

Mr. Putin’s master plan for Syria — promoted by his foreign minister, Sergey V. Lavrov — is clear: that the Western and Arab countries, which form the present anti-Islamic State coalition, should join forces with Mr. Assad, together with Kurdish and Iraqi troops; Iran, Hezbollah and Russia may also join this alliance. The coalition would obtain a formal mandate from the United Nations Security Council and then defeat the jihadist insurgency.

Russia would then bring Mr. Assad to the negotiating table and oversee a political transition that preserves his regime. Mr. Putin plans to address the United Nations General Assembly later this month about this plan.

In promoting a rapprochement between Russia and the West over the Islamic State, Mr. Putin hopes to rehabilitate himself, just as he did after Sept. 11. Back then, Mr. Putin convinced the West that the threat it faced in Afghanistan and elsewhere was the same as Russia faced in Chechnya. By doing so, Russia’s president was able to tamp down Western criticism of Russia’s brutality in Chechnya.

The Kremlin saw the West’s enthusiasm for cooperation as weakness. It led Mr. Putin to believe that he could act however he liked in Russia, and get away with it. That belief still prevails — but no longer applies only to Russia.