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Twisted homogeneous coordinate rings. (English) Zbl 0717.14001
Let \((X,\sigma)\) a pair consisting of a noetherian scheme \(X\) over a field \(k\) and an automorphism \(\sigma\) of \(X\). An invertible sheaf \({\mathcal L}\) on \(X\) is \(\sigma\)-ample if for every coherent sheaf \({\mathcal F}\) on \(X\), \(H^ q(X,{\mathcal L}\otimes {\mathcal L}^{\sigma}\otimes...\otimes {\mathcal L}^{\sigma^{n-1}}\otimes {\mathcal F})=0\) for \(q>0\) and sufficiently large \(n\). Let \(B=B(X,\sigma,{\mathcal L})\) be the twisted homogeneous coordinate ring of \(X\). On the other hand, let (\({\mathcal O}_ X\)-mod) be the category of quasi-coherent sheaves on \(X\), and (\(B\)-gr) the category of graded left modules over a graded ring \(B\). A graded module \(M=\oplus M_ n\) is called right bounded if \(M_ n=0\) for sufficiently large \(n\) and torsion if it is a direct limit of right bounded modules. Let (tors) be the subcategory of (\(B\)-gr) of torsion modules.
The aim of the paper is to prove that the categories (\({\mathcal O}_ X\)-mod) and (\(B\)-gr)/(tors) are naturally equivalent and that \(B\) is a finitely generated noetherian \(k\)-algebra. Finally, the case of a smooth surface is considered.
Reviewer: C.-P.Ionescu

14A15 Schemes and morphisms
14C20 Divisors, linear systems, invertible sheaves
18F99 Categories in geometry and topology
14A05 Relevant commutative algebra
Full Text: DOI
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