Coherent functors, with application to torsion in the Picard group

Details Coherent functors, with application to torsion in the Picard group Coherent functors, with application to torsion in the Picard group

1994-10-12

arxiv.org/abs/alg-geom/9410009v1

Mathematics

Algebraic Geometry

46 pages, AMS-LaTeX

Scientific paper

Let A be a commutative noetherian ring. Call a functor <> --> <> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When such a functor F in fact takes its values in <>, 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 show that the groups which can occur as the Picard group of a scheme of finite type over a finite field all have the form (finitely generated) + sum_{n=1}^infty F, where F is a finite p-group. Hard copy is available from the author. E-mail to jaffe@cpthree.unl.edu.

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