What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$?
Why (hopefully geometrically) should we not always have equality?
Notation. Let $I,J$ be two ideals of commutative ring. If $I=aR,J=bR$, denote the conductor of $J$ into $I$ by $(I:J)$ by $(a:b)$.
Definition. A commutative ring is quasicoherent if for any finite subset $ \left\{ a_1,\dots ,a_n,c \right\} \subset R$ the ideals $a_1R\cap \cdots \cap a_nR$ and $(0:c)$ are finitely generated.