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Inverse limits of algebras as retracts of their direct products

A. Laradji

2002
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Proceedings of the American Mathematical Society
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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
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... 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