Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various $\operatorname{Ext}^i$ groups, define projective dimensions and so on. In particular, one can define the graded global dimension $\operatorname{gr.gl.dim} A := \sup_{M \in M_A^g} \operatorname{pd} M$, where the projective dimension is computed in $M_A^g$, and the (ungraded) global dimension $\operatorname{gl.dim} A := \sup_{M \in M_A^u} \operatorname{pd} M$, where the projective dimension is computed in $M_A^u$.

For a graded $A$-module $M$ it is known that the projective dimensions in $M_A^g$ and in $M_A^u$ are actually the same. This immediately implies that $\operatorname{gr.gl.dim} A \leq \operatorname{gl.dim} A$. With a little more work, one can show that $\operatorname{gl.dim} A \leq 1 + \operatorname{gr.gl.dim} A$ (Theorem $II.8.2$ in Nastasescu, van Oystaeyen - Graded ring theory).

One can wonder whether is an example where the two notions actually differ (hence $\operatorname{gl.dim} A = 1 + \operatorname{gr.gl.dim} A$). Theorem $I.3.4$ in Nastasescu-van Oystaeyen ensures that $\operatorname{gr.gl.dim} A = \operatorname{gl.dim} A_0$ if $A$ is strongly graded over $\mathbb{Z}$, that is $1 \in A_i A_{-i}$ for all $i$. This provides us examples when the grading is over $\mathbb{Z}$, but of course it never applies with a grading over $\mathbb{N}$.

Hence my question:

is there an example of a ring $A$ graded over $\mathbb{N}$ such that $\operatorname{gl.dim} A = 1 + \operatorname{gr.gl.dim} A$?



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