Definition
Let \(G\) be an algebraic group over \(S\) and \(X\) a scheme over \(S\). An action of \(G\) on \(X\) is a morphism \(\alpha:G\times_S X\to X\) such that for every \(S\)-Scheme \(T\to S\), the map \(\alpha(T):G(T)\times X(T)\to X(T)\) is the action of \(G(T)\) on the set \(X(T)\).