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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

##### MSC:
 14A15 Schemes and morphisms 14C20 Divisors, linear systems, invertible sheaves 18F99 Categories in geometry and topology 14A05 Relevant commutative algebra
##### Keywords:
twisted homogeneous coordinate ring; torsion modules
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##### References:
 [1] Artin, M; Tate, J; Van den Bergh, M, Some algebras associated to automorphisms of elliptic curves, (), (to appear) · Zbl 0744.14024 [2] Grothendieck, A, Sur une note de mattuck-Tate, J. reine angew. math., 200, 208-215, (1958) · Zbl 0084.16904 [3] Griffiths, P; Harris, J, Principles of algebraic geometry, (), 741 [4] Murre, J.P, On contravariant functors from the category of preschemes over a field into the category of abelian groups, Publ. inst. math. hautes etudes sci., No. 23, (1964) · Zbl 0142.18402 [5] Nǎstäsescu, C; Van Oystaeyen, F, Graded ring theory, (), 16 · Zbl 0494.16001 [6] Serre, J.-P, Faisceaux algébriques cohérents, Ann. of math., 61, 197, (1955) · Zbl 0067.16201
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