A PROOF OF THE CHAN-YOUNG-ZHANG CONJECTURE

We prove the Chan-Young-Zhang conjecture on graded isolated singularities. Let k be a base field that is algebraically closed of characteristic zero. Fix an integer n ≥ 2. Let A = k−1[x0, . . . , xn−1] be the (−1)-skew polynomial ring, which is generated by {x0, . . . , xn−1} and subject to the relations xixj = (−1)xjxi for all i 6= j. Let G := Cn be the cyclic group of order n acting on A by permuting the generators of the algebra cyclically; namely, Cn is generated by σ = (0 1 2 · · · n− 1) of order n that acts on the generators by σxi = xi+1, ∀ i ∈ Zn := Z /nZ . The pertinency of a G-action on A [1, Definition 0.1] is defined to be p(A,G) := GKdim(A)−GKdim(A#G/(e0)), where (e0) is the ideal of the skew group algebra A#G generated by e0 := 1# 1 |G| ∑ g∈G g. To study noncommutative projective geometry, Ueyama gave the definition of a graded isolated singularity [8, Definition 2.2]. By [7, Theorem 3.10], A is a graded isolated singularity if and only if p(A,G) = GKdim(A). See [3, 2, 6, 5] for more examples of graded isolated singularities. In [4], Chan, Young and Zhang proved the following theorem on graded isolated singularities. Theorem 0.1. [4, Theorem 0.4] If either 3 or 5 divides n, then p(A,G) < GKdimA = n. Consequently, A is not a graded isolated singulartity. Based on the above theorem and [4, Theorem 0.2], they made the following conjecture. Conjecture 0.2. [4, Conjecture 0.5] Then A is a graded isolated singulartity if and only if n is not divisible by 3 or 5. Combining with the Theorem 0.1, to prove the conjective 0.2, we only need to prove the following theorem. Theorem 0.3. If n is not divisible by 3 or 5, then p(A,G) = GKdimA = n. As a consequence, A is a graded isolated singulartity.