A Beilinson-type theorem for coherent sheaves on weighted projective spaces
The aim of the paper is to extend to weighted projective spaces the theorem of \textit{A. A. Bejlinson} [Funct. Anal. Appl. 12, 214-216 (1979); translation from Funkts. Anal. Prilozh. 12, No. 3, 68-69 (1978; Zbl 0402.14006)] describing the bounded derived category \(D^b(\text{Coh} \mathbb{P}^n)\) of coherent sheaves on \(\mathbb{P}^n\). Let \(Q=(q_0, \dots,q_n)\) be a sequence of positive integers with \(\text{gcd}(q_0,\dots,q_n)=1\). The weighted projective space \(\mathbb{P}(Q)\) can be described as the quotient of \(\mathbb{P}^n\) by the action of the subgroup \(\mu_Q\) of \(GL(n+1)\) consisting of diagonal matrices with entries roots of unity of order \(q_0,\dots,q_n\), respectively. Let \(\pi:\mathbb{P}^n \to\mathbb{P} (Q)\) be the canonical projection. The author's idea is the following one: A careful analysis of the proof of Beilinson's theorem yields a version of this theorem for coherent \(\mu_Q\)-sheaves on \(\mathbb{P}^n\) and then one uses the fact that, for \({\mathcal F}\in\text{Coh} \mathbb{P} (Q)\), \(\pi^*{\mathcal F}\) is a \(\mu_Q\)-sheaf on \(\mathbb{P}^n\) and \({\mathcal F}\simeq (\pi_*\pi^* {\mathcal F})^{\mu_0}\). One has, however, to describe the sheaves \((\pi_*{\mathcal O}_{\mathbb{P}^n} (j))^\chi\) and \((\pi_*\Omega^j_{\mathbb{P}^n} (k))^\chi\) for every character \(\chi\) of \(\mu_Q\). The description of the later sheaves necessitates the introduction of some sheaves of ``logarithmic differentials'' on \(\mathbb{P}(Q)\).
scientific article; zbMATH DE number 1423602
scientific article