The Stacks project

Proof. The quotient $[X/G]$ is an algebraic stack by Criteria for Representability, Theorem 97.17.2. The smoothness of $[X/G]$ over $Y$ follows from the fact that smoothness descends under fppf coverings: Choose a surjective smooth morphism $U \to [X/G]$ where $U$ is a scheme. Smoothness of $[X/G]$ over $Y$ is equivalent to smoothness of $U$ over $Y$. Observe that $U \times _{[X/G]} X$ is smooth over $X$ and hence smooth over $Y$ (because compositions of smooth morphisms are smooth). On the other hand, $U \times _{[X/G]} X \to U$ is locally of finite presentation, flat, and surjective (because it is the base change of $X \to [X/G]$ which has those properties for example by Criteria for Representability, Lemma 97.17.1). Therefore we may apply Descent on Spaces, Lemma 74.8.4. $\square$