I'm reading Hatcher's book on algebraic topology. In Section 3.C, he proves as Theorem 3C.4 that if $A$ is a graded commutative associative Hopf algebra over a field of characteristic $0$ and $A^n$ is finitely generated for each $n$, then $A$ is isomorphic **as an algebra** to the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators. I emphasized the phrase **as an algebra** because I first assumed that this was a typo on his part, but indeed in his proof he only shows that $A$ has the right algebra structure. The coproduct is used to establish this, but he does not prove that $A$ has the same coproduct as the indicated tensor product.

Question : Must it? In other words, over a field of characteristic $0$ can the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators be given a Hopf-algebra structure other than the one coming from the tensor product?