Criteria for freeness in groups and valuated vector spaces

Hill, Paul

doi:10.1007/bfb0068193

Cited by 14 publications

(16 citation statements)

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“…Unfortunately, this theorem is erroneous. Its falsity was discovered first by F. Richman and E. A. Walker [5] and subsequently by E. White [6] as well as the present author [3]. The theorem was partially salvaged by Fuchs in [2].…”

mentioning

confidence: 90%

“…A valuated vector space W is said to be absolutely separable if it is separable in every containing space. Lemma 1 [3]. Any free space is absolutely separable.…”

mentioning

confidence: 98%

“…Recall that if Wis a subspace of V the valuation on the quotient space V/ W is given by \x + W\ = sup{|x + w\:w E. W) for any x G V. For convenience of proof that our example is valid, we shall employ the notion of a separable subspace which was introduced in [3]. Definition 1.…”

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confidence: 99%

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Quotients of valuated vector spaces

Hill¹

1981

Proc. Amer. Math. Soc.

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Abstract. This paper has three purposes. The first is to present a direct example of a quotient of an injective space that is not itself injective. The second is to demonstrate that Theorem 3 in [2] is false. An s -dense subspace S of a. free space F is not free by virtue of F/ S having only values that are cofinal with <•>. Finally, we furnish a valid proof, independent of the aforementioned Theorem 3, that a subspace V of a free space is free if V satisfies Fuchs' countability condition. . One consequence of Theorem 7 (if it were true) is that every valuated vector space has projective dimension at most one. The fact that Theorem 7 in [1] is false was discovered (by all authors mentioned above) indirectly through the discovery of spaces with projective dimension greater than 1. We shall give a direct counterexample, by exhibiting a subspace W of a product P of homogeneous spaces such that P/ W is not injective.The value of an element x is denoted herein by \x\. Recall that if Wis a subspace of V the valuation on the quotient space V/ W is given by \x + W\ = sup{|x + w\:w E. W) for any x G V. For convenience of proof that our example is valid, we shall employ the notion of a separable subspace which was introduced in [3]. Definition 1. A subspace W of V \% separable if \x + W\ = sup{|x + wn\} for some countable sequence {wn} of elements in W.Definition 2. A valuated vector space W is said to be absolutely separable if it is separable in every containing space.Lemma 1 [3]. Any free space is absolutely separable.The next result is of independent interest and is essential for our main result.Lemma 2. Any separable subspace of an injective valuated vector space is absolutely separable.

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confidence: 90%

“…A valuated vector space W is said to be absolutely separable if it is separable in every containing space. Lemma 1 [3]. Any free space is absolutely separable.…”

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confidence: 98%

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Quotients of valuated vector spaces

Hill¹

1981

Proc. Amer. Math. Soc.

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“…Every countably generated valuated space has a basis [1]. But in the uncountable case, even a subspace of a space with a basis need not have a basis; the author gave a necessary and sufficient condition for a subspace of a free valuated space to be again free in [3]. Although not every valuated space has a basis, it is true that every valuated space V has a basic subspace B.…”

Section: An Isomorphism Theorem For Valuated Vector Spacesmentioning

confidence: 99%

“…Following [3] and [4], we call a nice subspace of a free space an NSF-space. Equivalently, an NSF-space is a nice subspace of a valuated vector space that has a basis.…”

Section: An Isomorphism Theorem For Valuated Vector Spacesmentioning

confidence: 99%

An isomorphism theorem for valuated vector spaces

Hill¹

1983

Proc. Amer. Math. Soc.

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The following isomorphism theorem is proved for valuated vector spaces. Let N N and N ′ N’ be nice subspaces of free valuated vector spaces F F and F ′ F’ , respectively. If N N and N ′ N’ have isomorphic basic subspaces and if the quotient spaces F / N F/N and F ′ / N ′ F’/N’ are isomorphic, there exists an isomorphism from F F onto F ′ F’ that maps N N onto N ′ N’ and induces the given isomorphism on the quotient spaces. In particular, N N and N ′ N’ are isomorphic.

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