ScalingStacks

Definition 2.84. A character on $A_{Z}$ is a homomorphism of $k$-algebra from $A_{Z}$ to some valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ over $(k,\lvert\mathord{\cdot}\rvert_{k})$. Two characters $\chi_{1}:A_{Z}\to K_{1}$ and $\chi_{1}:A_{Z}\to K_{1}$ are called equivalent if there exists a $k$-algebra homomorphism $\chi_{3}:A_{Z}\to K_{3}$ and norm preserving $k$-algebra homomorphisms $i_{1}:K_{1}\to K_{3}$ and $i_{2}:K_{2}\to K_{3}$ satisfying $\chi_{3}=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$.