Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus\_{i=1}^{\infty} V^{\otimes i}
$$
and the quotients
$$
A\_W = T(V)/\langle W \rangle, ~~~~~~~~~~~ A\_U = T(V)/\langle U \rangle
$$
by the ideals generated by $W$ and $V$ respectively. The quotients are again $\mathbb{Z}$-graded algebras (in fact quadratic algebras) and we have a surjective map from every homogeneous subspace of $A\_U$ to every homogeneous subspace of $A\_V$. At degree $2$ this is clearly not injective. However at higher degrees I would conjecture that it can be an isomorphism although I cannot see an example. I am most interested in the case where $A\_U$ is a finite-dimensional algebra. In this can it happen that the non-zero homogeneous subspace of $A\_U$ of highest order can have the same degree as the non-zero homogeneous subspace of $A\_W$ of highest order?



A question about surjective maps between quadratic algebras
There are plentiful examples of $m$-generated quadratic algebras $A$ with $A\_3 = 0$, and dimension of relation subspace $R$ being as low as $m^2/2$. One of those algebras is $$\Bbb k\langle x\_1, y\_1, \dots, x\_n, y\_n \rangle / (x\_i y\_j, x\_i x\_j - y\_i y\_j).$$ Moreover, if $\operatorname{dim} R \geq 3m^2/4$, then for very generic (i. e. lying in intersection of countably many certain Zariski opens in Grassmanian) choice of $R$ corresponding quadratic algebra will have $A\_3 = 0$ and be Koszul.


Obviously, any proper quadratic quotient of those will give you an example of epimorphism of quadratic algebras such that their Hilbert series only differ in degree 2. You can take direct sum with your favourite quadratic algebra to have something in degrees $\geq 3$ if you want.


UPD: I need to do a due diligence and add reference. I can wholeheartedly recommend to anyone interested in quadratic algebras and combinatorial algebra in general to read small book "Quadratic algebras" by A. Polishchuk and L. Positselski. <https://bookstore.ams.org/ulect-37>


