Exercise 111.26.9. Let $k$ be a field and let $A = k$ and $B = k[x, y]/(x^2, xy)$ with grading determined by $\deg (x) = 2$ and $\deg (y) = 3$. Let $\varphi (M) = \dim _ k(M)$. Compute the Hilbert function of $B$ as a graded $k$-module. Is there a Hilbert polynomial in this case?
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