Baire's category theorem and prime avoidance in complete local rings (Q798373)

Let A be a complete (commutative Noetherian) local ring (with identity) having maximal ideal \({\mathfrak m}\). Let (\({\mathfrak p}_ i)_{i=1,2,...}\) be a countable family of prime ideals of A. In Proc. Camb. Philos. Soc. 72, 369-373 (1972; Zbl 0242.13018), \textit{L. Burch} showed that, if none of the \({\mathfrak p}_ i\) is equal to \({\mathfrak m}\), then there is an element of \({\mathfrak m}\) which is not contained in any \({\mathfrak p}_ i\). One of the purposes of the present note is to show that Baire's category theorem can be used to prove the above and other related ''countable prime avoidance'' results: it is shown that, for \(x\in A\) and \({\mathfrak a}\) an ideal of A, (i) if \({\mathfrak a}\subseteq\cup^{\infty}_{i=1}{\mathfrak p}_ i,\) then \({\mathfrak a}\subseteq {\mathfrak p}_ j\) for some \(j\geq 1\), and (ii) if \(Ax+{\mathfrak a}\varsubsetneq\cup^{\infty}_{i=1}{\mathfrak p}_ i,\) then there exists \(r\in {\mathfrak a}\) such that \(x+r\not\in\cup^{\infty}_{i=1}{\mathfrak p}_ i.\) The paper also shows that the ''countable prime avoidance'' property automatically holds in a (not necessarily complete) (commutative Noetherian) local ring which has uncountable residue field. In the second part of the paper, some of the above results are applied to a countably generated A-module M for which \(M\neq {\mathfrak m}M\) to show that at least some of the theory of grade and depth for finitely generated A- modules can be extended to such an M: it is shown that, if \({\mathfrak a}\) is a proper ideal of A, then all maximal M-sequences in \({\mathfrak a}\) have the same length, namely the least integer i such that \(Ext^ i_ A(A/{\mathfrak a},M)\neq 0.\) The paper concludes with an application to countably generated big Cohen-Macaulay A-modules.