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