Sigismund 3 and false Dmitry – allies or opponents?

After the direct line of the Rurikovichs was interrupted in Russia, B. Godunov ascended the throne. At the same time, False Dmitry appears, who announces that he is a miraculously saved heir. But he needed support to get the throne. He flees to Poland, where Sigismund III promises to help him get the throne if he gives up part of the Russian lands. False Dmitry agrees. But after he becomes king he refuses to fulfill the terms of the contract. The king besieges Smolensk. So, each of them pursued their own interests and was an ally only for a short time.