# Is associated graded algebra $\mathrm{gr}(k[x_1, \ldots, x_n]/I)$ isomorphic as a vector space to $k[x_1, \ldots, x_n]/I$?

Let $$A=k[x_1, \ldots, x_n]$$ be the polynomial ring generated by $$x_1, \ldots, x_n$$. Let $$I$$ be an ideal of $$k[x_1, \ldots, x_n]$$ (it is possible that $$I$$ is not homogeneous).

The algebra $$A$$ is a graded algebra: $$A = \oplus_{i \ge 0} A_i$$, where $$A_i$$ consists of degree $$i$$ homogeneous polynomials.

The algebra $$A/I=k[x_1, \ldots, x_n]/I$$ is a filtered algebra with the filtration $$F_i(A/I) = (F_i(A)+I)/I$$, where $$F_i(A)=\oplus_{j \le i} A_j$$.

Is associated graded algebra $$\mathrm{gr}(k[x_1, \ldots, x_n]/I)=\mathrm{gr}(A/I)=\oplus_{i \ge 0} F_i(A)/(F_{i-1}(A) + F_i(A) \cap I)$$ isomorphic as a vector space to $$k[x_1, \ldots, x_n]/I$$?

Thank you very much.

Observe that $$\mathrm{gr}(A/I)$$ is also a filtered algebra, with $$F_i(\mathrm{gr}(A/I)) = \oplus_{i=0}^r~\mathrm{gr}(A/I)_i$$.
Further observe that $$\dim F_i(\mathrm{gr}(A/I) = \sum_{i=0}^r \dim \mathrm{gr}(A/I)_i = \sum_{i=0}^r (\dim F_i(A/I) - \dim F_{i-1}(A/I)) = \dim F_i(A/I).$$
Therefore $$\mathrm{gr}(A/I)$$ is isomorphic to $$A/I$$ as a vector space.