1994/10/12 by David B. Jaffe, Jaffe, David B.
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.alg-geom/9410009
openalex publication_date 1994/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ↦ M tensorA 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) + sumn=1infty F, where F is a finite p-group. Hard copy is available from the author. E-mail to [email protected].