On the $\phi$-Weak Global Dimensions and $\phi$-Global Dimensions

Let $R$ be a $\phi$-ring and put $N=Nil(R)$. We determine the $\phi$-weak global dimension of every $\phi$-Prüfer ring. Its only possible values are $0$, $1$, and $\mathrm{\infty }$: the value is $0$ when $R\mathrm{/}N$ is a field, it is $1$ when $R$ is strong and $R\mathrm{/}N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\to \phi (R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R\mathrm{/}N$, and its self-Tor groups satisfy $$To{r}_{2n}^R(K,K)=0(n\ge 1),To{r}_{2n+1}^R(K,K)\mathrm{\ne }0(n\ge 0)\mathrm{.}$$ It follows that $f{d}_{R}K=\mathrm{\infty }$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $\phi$-global dimension of $\phi$-Dedekind rings. For strong $\phi$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\to R\mathrm{/}N$ splits. If $D=R\mathrm{/}N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then $$R\cong T{\times }_{Q}D\mathrm{.}$$ Using Ferrand’s flat patching theorem and Milnor’s projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\to D$, satisfies $$f{d}_R(M)=f{d}_D(M),p{d}_R(M)=p{d}_D(M)\mathrm{.}$$ Consequently, for every strong $\phi$-ring, $$\phi -wgd(R)=wgld(R\mathrm{/}N),\phi -gld(R)=gld(R\mathrm{/}N)\mathrm{.}$$