| Coherent functors, with application to torsion in the Picard group (2007) | |||||||||||||||
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| Let A be a commutative noetherian ring. We investigate a class of functors from #commutative A-algebras# to #sets#, which we call coherent. When such a functor F in fact takes its values in #abelian groups#, we show that there are only finitely many prime numbers p such that p F(A) is infinite, and that none of these primes are invertible in A. This (and related statements) yield information about torsion in Pic(A). For example, if A is of finite type over Z, we prove that the torsion in Pic(A) is supported at a finite set of primes, and if p Pic(A) is infinite, then the prime p is not invertible in A. These results use the (already known) fact that if such an A is normal, then Pic(A) is finitely generated. We obtain a parallel result for a reduced scheme X of finite type over Z. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field. Coherent functors (introductory remarks) Let us say that an A-functor is a functor from the c... | |||||||||||||||
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