Coherence and weak global dimension of R[[X]] when R is von Neumann regular

Abstract

dimension of R. Very little is known about what can occur except that the weak global dimension must increase by at least one and does increase by exactly one when R[[X]] is coherent. This paper attempts to cast some light on this problem and the closely related question of the stability of coherence under the formation of the power series ring. It is devoted to determining necessary and sufficient conditions on a commutative (von Neumann) regular ring R for the power series ring R[[X]] to b e coherent (equivalently, semihereditary) and also conditions for R[[X]] t o h ave weak global dimension one. Surprisingly, it turns out that R[[X]] h as weak global dimension one precisely when R[[X’j] is a B&out ring so that property is characterized as well. Each of these properties is characterized in several ways both in terms of internal conditions on R and conditions that involve the category of R-modules. Perhaps the conditions which can be most readily verified for a specific ring are expressed in terms of a natural partial order z< which is defined on R by a < b if and only if ab = a2 for a, b in R. For example, it is shown that R[[X]] is coherent if and only if every countable subset of R that forms a chain in the partial order < on R has a least upper bound in R. As further illustrations of our results we mention that R[[Xj] h as weak global dimension one precisely in case R satisfies either of two limited forms of self-injectivity and also in case R satisfies a restricted type of algebraic compactness. These results permit us to give an example that shows R[[X]] can have weak global dimension one without being coherent even when R is a Boolean ring. This settles a question raised by Jensen [7]. Another example is included to illustrate the added complexity that occurs in arbitrary regular rings as compared with Boolean rings. This example also provides a pair of regular rings R contained in S, and hence a faithfully flat extension, such that R[[X]] is not pure in S[[X]]. Thus it gives a negative