Global dimension.

Question

1. What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$?

2. What is the global dimension of $\mathbb{Z}/p\mathbb{Z}$?

Answer 1

Here are three results which in principle reduce the computation of global dimension to that of Krull dimension:

a) The global dimension of a noetherian ring $A$ is the supremum of the global dimension at its localizations at its maximal ideals: $ \text {gldim} A=\text {sup}_{\mathfrak m \in\text {Specmax }A} \; \text {gldim} {A_\mathfrak m } $
b) For a regular noetherian local ring the global dimension equals the Krull dimension: $ \text {gldim} A=\text {dim} A$
c) For a non-regular local noetherian ring $ \text {gldim} A=\infty$

Your questions are then immediately solved (if you notice that $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$ is of the form $\mathbb Z/(n))$:$$\quad \text {gldim} \: \mathbb Z_{(p)}=1$$and$$ \quad \text {gldim} \;\mathbb Z/(n)=0 \quad\text {or}\quad\text {gldim} \;\mathbb Z/(n)=\infty$$ according as $n$ is square-free or not.