Suppose that $R$ is a local ring. Then $GL_2(R)$ acts on the space of binary cubic forms $$p(x,y)=ax^3+bx^2y+cxy^2+dy^3, \quad a,b,c,d\in R,$$ by $$g\cdot p(x,y)=p((x,y)g).$$ My question is how to show that the action is faithful.
I'm reading the paper by Gross and Lucianovic, ON CUBIC RINGS AND QUATERNION RINGS. The case when $R$ is a PID is easy, but I don't know how to proceed when $R$ is a local ring.
