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Nonsplitting Sequences of Value Groups

Joe L. Mott
Proceedings of the American Mathematical Society
Vol. 44, No. 1 (May, 1974), pp. 39-42
DOI: 10.2307/2039223
Stable URL: http://www.jstor.org/stable/2039223
Page Count: 4
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Nonsplitting Sequences of Value Groups
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Abstract

If K is the quotient field of an integral domain D, then the value group VK(D) of D in K is the group K*/U(D), partially ordered by D*/U(D), where U(D) denotes the group of units of D. This note shows that if the sequence \begin{equation*} \tag{1} \{1\} \rightarrow G \rightarrow H \rightarrow J \rightarrow \{1\}\end{equation*} is lexicographically exact and if H is lattice ordered, then there is a Bezout domain B and a prime ideal P of B such that VK(B) = H, VK(BP) = J, and VK(B/P) = G, where k denotes the residue field of BP. Moreover, B is the direct sum of B/P and P, and BP = k + P. In particular, the sequence (1) need not split, even with somewhat stringent restrictions on the integral domain B. This gives a negative answer to a question posed by R. Gilmer.

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