Regular ℤ-graded local rings and graded isolated singularities

In this note we first study regular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded local rings. We characterize commutative noetherian regular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded algebra generated in degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.