Hereditary localization of polynomial rings (Q1176653)

Let \(R\) be a commutative ring and \(x\) an indeterminate over \(R\). Let \(U\) denote the set of monic polynomials in \(R[x]\) and let \(V\) denote the set of polynomials in \(R[x]\) with content \(R\). Let \(R\langle x\rangle\) and \(R(x)\) denote the localization of \(R[x]\) at \(U\) and \(V\) respectively. This paper is concerned with the behavior of the global dimensions of \(R\langle x\rangle\) and \(R(x)\). An extension \(S\) of \(R\) is said to ascend global dimension if \(\hbox{gl.}\dim(R)=n\) implies that \(\hbox{gl.}\dim(S)\leq n\); it is said to descend global dimension if \(\hbox{gl.}\dim(S)=n\) implies that \(\hbox{gl.}\dim(R)\leq n\). It is shown that the extensions \(R\langle x\rangle\) and \(R(x)\) of \(R\) both ascend and descend hereditarity, that is, global dimension equal to one. Furthermore for an \(\aleph_ 0\)-Noetherian ring \(R\) these extensions descend any finite global dimension. An example is provided of a class of local, stably coherent, \(\aleph_ 0\)-Noetherian rings \(R\) of global dimension equal to two, for which the extension \(R\langle x\rangle\) does not descend global dimension.