# Ring with different graded and ungraded global dimensions

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 $$\operatorname{gl.dim} A = 1 + \operatorname{gr.gl.dim} A$$?