Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain.
Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a canonical divisor on $\mathbf{P}^1_A$.
How do I do this?
I know how to do this when $A=k$ is a field. In this case, the identity morphism $\mathbf{P}^1_k\to \mathbf{P}^1_k$ is a rational function $f$ and div $df$ is a canonical divisor. One easily computes that div $df = -2 [\infty]$.
I might actually be wrong...In this case, how do I compute a canonical divisor for $\mathbf{P}^1_A$. What if $A$ is a Dedekind domain?