If you take $A=\mathcal{O}_K$, the invertible $A$-modules are exactly the non-zero fractional ideals of $A$, so I guess we can reformulate the exercise to be
There is a bijection of sets $\{(n+1)$-tuples of elements of A such that $\exists i: a_i\neq 0 \}$ modulo equivalence, where equivalence is multiplication by a non-zero element of $A$K$and$\{A$-valued points of$\mathbb{P}^n_A\}$Thus, for$\mathcal{O}_K$, the "classical" definition of points of projective$n$-space coincides with the definition of$\mathcal{O}_K$-valued points of $\mathbb{P}^n_{\mathcal{O}_K}$. 2 I also forgot "fractional" in the above... Thanks for your answers!! If you take$A=\mathcal{O}_K$, the invertible$A$-modules are exactly the non-zero fractional ideals of$A$, so I guess we can reformulate the exercise to be There is a bijection of sets$\{(n+1)$-tuples of elements of A such that$\exists i: a_i\neq 0 \}$modulo equivalence, where equivalence is multiplication by a unitnon-zero element of$A$and$\{A$-valued points of$\mathbb{P}^n_A\}$Thus, for$\mathcal{O}_K$, the "classical" definition of points of projective$n$-space coincides with the definition of$\mathcal{O}_K$-valued points of $\mathbb{P}^n_{\mathcal{O}_K}$. 1 Thanks for your answers!! If you take$A=\mathcal{O}_K$, the invertible$A$-modules are exactly the non-zero ideals of$A$, so I guess we can reformulate the exercise to be There is a bijection of sets$\{(n+1)$-tuples of elements of A such that$\exists i: a_i\neq 0 \}$modulo equivalence, where equivalence is multiplication by a unit and$\{A$-valued points of$\mathbb{P}^n_A\}$Thus, for$\mathcal{O}_K$, the "classical" definition of points of projective$n$-space coincides with the definition of$\mathcal{O}_K$-valued points of $\mathbb{P}^n_{\mathcal{O}_K}\$.