Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what is $f^*\mathcal{N}$?

Question

Say $f : \mathbb{A}^2 \rightarrow \mathbb{A}$ is projection on the first coordinate. Given a sheaf $\mathcal{N}$ of $\mathcal{O}_Y$-modules defined by a finitely generated $\mathbb{C}[x]$-module, what is $f^*\mathcal{N}$? Does it come from a $\mathbb{C}[x, y]$-module?

Answer 1

I assume by $Y$ you mean $\mathbb A^1 = Spec(\mathbb C [x])$. Then the answer to your last question is yes, which is the content of Proposition 5.8, Chapter II in Hartshorne's book.

(just as Ehsan Kermani says in his comment)

Since you are dealing with affine schemes, you can identify $\mathcal N$ with its set of global sections $N=\Gamma(\mathbb A^1, \mathcal N)$. Then, again as Ehsan says, to compute the pullback, you just tensor with $k[x,y]$, where now $k[x,y]$ acts on $N$ as follows: $x$ acts as before, but $y \cdot n = 0$ for all $n$.