Regular ℤ-graded local rings and graded isolated singularities

In this note we first study regular Z Z -graded local rings. We characterize commutative noetherian regular Z Z -graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative Z 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 N N -graded algebra generated in degree 1 1 is a graded isolated singularity if and only if its associated noncommutative 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.