Inverse limits of algebras as retracts of their direct products

A. Laradji
2002 Proceedings of the American Mathematical Society  
Inverse limits of modules and, more generally, of universal algebras, are not always pure in corresponding direct products. In this note we show that when certain set-theoretic properties are imposed, they even become direct summands. Given a direct system {M i } i∈I of modules, it is well known that lim − → M i is a pure quotient of the direct sum i∈I M i . In contrast, the dual statement that inverse limits are pure submodules of corresponding direct products is not always true: For each
more » ... true: For each prime number p, we can construct a descending chain {A n } n∈N of divisible abelian groups whose intersection A is isomorphic to Z/pZ (see [2, Exercise 6, p. 101]). Since divisibility is inherited by pure subgroups and direct products and since A is not divisible, it follows that the inverse limit A of the divisible groups A n is not pure in n∈N A n . However, as we shall show in this note, when certain set-theoretic conditions are imposed on an inverse system of modules, the inverse limit is a direct summand of the corresponding direct product. This is motivated by the following observation: Let p be a prime number and let J p be the p-adic group lim ← − Z/p n Z. As each Z/p n Z is finite, J p is linearly, and hence algebraically compact. (See [1] and [2].) Since, as can easily be proved, J p is pure in n Z/p n Z, it follows that the canonical monomorphism 0 → lim ← − Z/p n Z → n Z/p n Z splits. The purpose of this note is to generalize this result in both set-theoretic and universal algebraic directions. We refer to [4] and [3] for the various undefined notions used here from the theory of large cardinals and universal algebra, respectively. Recall that a tree is a poset (T, <) such that for each t ∈ T the set {s ∈ T : s < t} of the predecessors of t is well ordered by <. A subalgebra B of an algebra A is a retract of A if there exists a homomorphism g : A → B whose restriction to B is the identity on B; such a g is called a retraction. A directed set {I; ≤} is λ-directed for some infinite cardinal λ, if every subset of I of size less than λ has an upper bound in I. First, we need
doi:10.1090/s0002-9939-02-06666-2 fatcat:ce6koxrpqjbddbbtan54jauype