Coherent functors, with application to torsion in the Picard group

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 pF (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 category of commutative A-algebras to ≪sets≫. Some such A-functors have additional structure: they are actually functors from ≪commutative A-algebras≫ to ≪groups≫. We refer to such functors as group-valued A-functors. We will also consider A-functors F such that F (B) is a B-module for every B; these module-valued A-functors are discussed later in the introduction. For now, all A-functors which we consider will be treated as set-valued functors. An A-functor is coherent if it may be built up as an iterated finite limit of functors of the form M , given by M(B) = M ⊗A B, where M is a finitely generated A-module. We do not know if every coherent functor may be expressed as a finite limit of such functors M . However, the analogous question regarding module-valued functors is answered affirmatively below. The idea of coherent functor was originally devised by Auslander [5], in a somewhat different setting; his notion of coherence applied to functors from an abelian category to ≪abelian groups≫. Later Artin [2] transposed Auslander’s notion to a setting closer to that given here. Artin also raised a question about coherence of higher direct images as functors. This question is considered in §7. If an A-functor is representable by a commutative A-algebra of finite type, then it is coherent. There are many examples of non-representable A-functors which are coherent. For example, if M is a finitely generated A-module, then B 7→ AutB−mod(M ⊗A B) defines a coherent A-functor. More examples may be found in §4. A module-valued A-functor is an (abelian group)-valued A-functor F , together with the following additional structure: for each commutative A-algebra B, F (B) has the structure of a B-module, such that 1991 Mathematics Subject Classification. Primary: 14C22, 18A25, 14K30, 18A40.