I asked a question about linear maps between free modules. I have a related question.
Let $S=k[x_1, \ldots, x_n]$ where $k$ is a field. Then $S$ is a graded ring with usual grading given by $\deg x_i = 1$. By definition, $S(a)$ is a graded module defined by $S(a)_d = S_{a+d}$. What is the basis of $S(a)$? For example, what is the basis of $S(-3)$?
Thank you very much.
The basis of $S(a)$ is formed by all monomials $x^i = x_1^{i_1}\cdots x_n^{i_n}$, but the degree of $x^i$ is equal to $i_1 + \dots + i_n - a$.