Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities

Abstract: In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. The characterization by the length of (homogeneous) regular sequences fails in the graded case in general. Then, we characterize graded isolated singularity for commutative $\mathbb{Z}$-graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine $\mathbb{N}$-graded algebra generated in degree $1$ is a graded isolated singularity if and only if its associated projective scheme is smooth; if and only if the category of coherent sheaves on its projective scheme has finite global dimension, which are known in literature.