When is the quotient of an algebra by a polynomial algebra a free module?

Question

Let $A$ be a $k$-algebra and let $V$ be a subalgebra of $A$ that is a polynomial algebra. Is $A$ a free module over $V$?

One example is when $A=\mathbb{C}[x_1, x_2, \ldots x_n]$ and $V$ is the ring of symmetric polynomials in $n$ variables. Can you prove that $A$ is free over $V$?