Projective coordinates over a local ring ($\mathrm{Hom}(\mathrm{Spec}A,{\mathbb{P}}^n)=\{(a_i):\exists i,a_i\in A^{\times }\}/A^{\times }$)

For a local ring $A$, every map from $\mathrm{Spec}A$ into projective space lands in one standard chart: the inverse image of a chart containing the closed point is the whole spectrum. This gives homogeneous coordinates with at least one unit entry, unique up to a common unit. For a nonlocal ring, coordinates may instead merely generate the unit ideal, and general maps can involve a nontrivial invertible sheaf quotient.