Coherent functors, with application to torsion in the Picard group |
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Authors: | David B Jaffe |
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Institution: | Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323 |
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Abstract: | Let be a commutative noetherian ring. We investigate a class of functors from commutative -algebras to sets , which we call coherent. When such a functor in fact takes its values in abelian groups , we show that there are only finitely many prime numbers such that is infinite, and that none of these primes are invertible in . This (and related statements) yield information about torsion in . For example, if is of finite type over , we prove that the torsion in is supported at a finite set of primes, and if is infinite, then the prime is not invertible in . These results use the (already known) fact that if such an is normal, then is finitely generated. We obtain a parallel result for a reduced scheme of finite type over . We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field. |
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Keywords: | Coherent functor representable functor Picard group |
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