Is $i^*(\mathcal O_{\mathrm P_S^m}(1))$ an invertible $\mathcal O_X$-module?

Question

Let $X$ be an $S$-scheme.

For any immersion $i: X\to \mathrm P_S^m$ which is an $S$-morphism, is $i^*(\mathcal O_{\mathrm P_S^m}(1))$ an invertible $\mathcal O_X$-module?

Answer 1

To expand on Mohan's comment, we say a sheaf $\mathcal{L}$ on $Y$ is invertible if there exists a sheaf $\mathcal{N}$ on $Y$ such that $\mathcal{L} \otimes \mathcal{N} \cong \mathcal{O}_Y$. Pulling back, we have $$f^*(\mathcal{L} \otimes \mathcal{N}) \cong f^*\mathcal{L} \otimes f^* \mathcal{N} \cong \mathcal{O}_X \cong f^* \mathcal{O}_Y $$ where the first isomorphism is by the fact that tensor commutes with pullback, and the last isomorphism is by the fact that the pullback of the structure sheaf on $Y$ is the structure sheaf on $X$. So $f^* \mathcal{L}$ is invertible.

Since $\mathcal{O}(1)$ is invertible ($\mathcal{O}(1) \otimes \mathcal{O}(-1) \cong \mathcal{O}$), $i^*(\mathcal{O}(1))$ is too.