I am trying understand associated graded algebras.
Let $A=k[x,y]$ and $I=\langle xy^2, x^3-y^2 \rangle$. Let $R=A/I$. In the webpage, it is said that $\mathrm{gr}(R) = A/(x^4, y^2)$.
Let $A=k[x_1, \ldots, x_n]$ and ``$>$'' a monomial order on $A$. For a polynomial $p$, denote by $in_>(p)$ the ideal the term of $p$ with the largest monomial. For an ideal $I$ of $A$, the initial ideal of $I$ is defined as $in_{>}(I) = \langle in_{>}(p) \mid p \in I \rangle$.
I think that $gr(R)$ is isomorphic to $A/in(I)$ as a vector space, where $in(I)$ is the initial ideal of $I$. Is this correct? I obtain that $gr(R)$ is isomorphic to $k[x,y]/(xy^2, x^3, y^4)$ as a vector space. Is this correct? Thank you very much.