Nonsplitting sequences of value groups

Abstract:

If $K$ is the quotient field of an integral domain $D$, then the value group ${V_K}(D)$ of $D$ in $K$ is the group ${K^ \ast }/U(D)$, partially ordered by ${D^ \ast }/U(D)$, where $U(D)$ denotes the group of units of $D$. This note shows that if the sequence \[ (1)\quad \{ 1\} \to G \to H \to J \to \{ 1\} \] 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 ${V_K}(B) = H,{V_K}({B_P}) = J$, and ${V_k}(B/P) = G$, where $k$ denotes the residue field of ${B_P}$. Moreover, $B$ is the direct sum of $B/P$ and $P$, and ${B_P} = k + P$. 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.