# The variety associated to a polynomial ring with a particular grading

We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety.

Suppose $k[x_1,x_2, x_3, x_4]$ is a polynomial ring over $k=\bar{k}$.

Let $R = k[x_1, x_2]$ and consider the graded module

$$S = R \oplus (R x_1 x_3 \oplus R x_4) \oplus (R x_1 x_3 x_ 4 \oplus R x_4^2 \oplus R x_1^2 x_3^2) \oplus \ldots .$$

What is $Proj(S)$?

Any comment or feedback is appreciated.



Note that $R x_1 x_3 = \{ a x_1 x_3: a \in R\} = \{ b x_3: b \in R x_1\}$. 

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If I interpret well, the ring $S$ coincide with $k[x_1, x_2, x_1 x_3, x_4]$ with grading $\deg x_1 = \deg x_2 = 0$ and $\deg x_1 x_3 = \deg x_4 = 1$. It is quite clear that $x_1 x_3$ is trascendent over $k[x_1, x_2, x_4]$, hence $S$ is a polynomial ring over $R = k[x_1, x_2]$ with two indeterminates, therefore $$\mathrm{Proj}(S) = \mathbb{P}^1_R = \mathbb{P}^1_k \times_k \mathrm{Spec}(R) = \mathbb{P}^1_k \times_k \mathbb{A}^2_k.$$

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