Noncommutative graded domains with quadratic growth
Identifieur interne : 000345 ( Istex/Curation ); précédent : 000344; suivant : 000346Noncommutative graded domains with quadratic growth
Auteurs : M. Artin [États-Unis] ; J. T. Stafford [États-Unis]Source :
- Inventiones mathematicae [ 0020-9910 ] ; 1995-12-01.
English descriptors
- KwdEn :
- Algebra, Algebraic curve, Analogue, Artin, Asymptotic structure, Automorphism, Birational, Centre, Coherent sheaf, Coherent sheaves, Commutative, Commutative ring, Corollary, Division ring, Divisor, Divisor sequence, Domain, Exact sequence, Field extension, Finite codimension, Finite module, Finite orbits, Finite order, Finitely, Function field, Global sections, Goldie, Homogeneous element, Homogeneous elements, Ideal sheaf, Inequality, Infinite orbit, Infinite orbits, Infinite order, Integer, Invertible, Invertible sheaf, Invertible sheaves, Isomorphic, Last paragraph, Lemma, Module, Morphism, Next lemma, Next result, Noetherian, Noetherian domain, Noncommutative, Noncommutative analogue, Nonzero, Nonzero element, Nonzero right, Notation, Opposite inequality, Other hand, Other words, Positive degree, Prime ideals, Prime ring, Projective, Projective curve, Projective scheme, Quadratic, Quadratic growth, Quotient, Quotient category, Quotient ring, Regular element, Right module, Right noetherian, Rmax, Sheaf, Simple artinian ring, Singular point, Stable image, Stafford, Subalgebra, Subring, Subscheme, Suffices, Surjective, Theorem, Transcendence degree, Vector space, Veronese, Veronese ring, Veronese rings, Veronese sequence, Veronese subring.
- Teeft :
- Algebra, Algebraic curve, Analogue, Artin, Asymptotic structure, Automorphism, Birational, Centre, Coherent sheaf, Coherent sheaves, Commutative, Commutative ring, Corollary, Division ring, Divisor, Divisor sequence, Domain, Exact sequence, Field extension, Finite codimension, Finite module, Finite orbits, Finite order, Finitely, Function field, Global sections, Goldie, Homogeneous element, Homogeneous elements, Ideal sheaf, Inequality, Infinite orbit, Infinite orbits, Infinite order, Integer, Invertible, Invertible sheaf, Invertible sheaves, Isomorphic, Last paragraph, Lemma, Module, Morphism, Next lemma, Next result, Noetherian, Noetherian domain, Noncommutative, Noncommutative analogue, Nonzero, Nonzero element, Nonzero right, Notation, Opposite inequality, Other hand, Other words, Positive degree, Prime ideals, Prime ring, Projective, Projective curve, Projective scheme, Quadratic, Quadratic growth, Quotient, Quotient category, Quotient ring, Regular element, Right module, Right noetherian, Rmax, Sheaf, Simple artinian ring, Singular point, Stable image, Stafford, Subalgebra, Subring, Subscheme, Suffices, Surjective, Theorem, Transcendence degree, Vector space, Veronese, Veronese ring, Veronese rings, Veronese sequence, Veronese subring.
Abstract
Abstract: Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.
Url:
DOI: 10.1007/BF01231444
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<term>Other words</term>
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<term>Subscheme</term>
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<term>Theorem</term>
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<term>Asymptotic structure</term>
<term>Automorphism</term>
<term>Birational</term>
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<term>Coherent sheaf</term>
<term>Coherent sheaves</term>
<term>Commutative</term>
<term>Commutative ring</term>
<term>Corollary</term>
<term>Division ring</term>
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<term>Divisor sequence</term>
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<term>Exact sequence</term>
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<term>Next result</term>
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<term>Noncommutative analogue</term>
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<term>Nonzero element</term>
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<term>Notation</term>
<term>Opposite inequality</term>
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<term>Quotient ring</term>
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<term>Sheaf</term>
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<term>Theorem</term>
<term>Transcendence degree</term>
<term>Vector space</term>
<term>Veronese</term>
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<front><div type="abstract" xml:lang="en">Abstract: Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.</div>
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