Let R be a graded (associative, unital) ring. If R is left graded hereditary (i.e. its left graded global dimension is 0 or 1), does it follow that R is left hereditary (i.e. its left global dimension is 0 or 1)?
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1$\begingroup$ The title of this good question is a good example of the ambiguity of “any” in mathematical English. I think it would be better to use “every” — I first read the title as asking if some graded hereditary ring is hereditary. $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Oct 3, 2023 at 12:45
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$\begingroup$ @Carl-Fredrik Nyberg Brodda Thank you for this comment. I have changed the title. $\endgroup$– RalleCommented Oct 3, 2023 at 14:56
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No, consider the graded ring $R_{\ast}=\mathbb{Z}[x,x^{-1}]$ with $|x|=1$. Then the functor $M\mapsto M_0$ gives an equivalence from graded $R_{\ast}$-modules to abelian groups, so $R_{\ast}$ is hereditary in the graded sense. However, the corresponding ungraded ring is not hereditary: for example, the inhomogeneous ideal generated by $(2,x-1)$ is not a free module.
answered Oct 3, 2023 at 10:42