Math. Z. 180, 1 - 9 (1982) Nlathematische Zeitschrift 9 Springer-Verlag 1982 Homological Dimension and Stationary Sets Paul C. Eklof* Mathematics Department, Bedford College, University of London, Regent's Park, London NW1 4NS, England Current address: Mathematics Department, Universityof California, Irvine, CA 92717, U.S.A. Introduction Throughout this paper R will denote a ring with 1, and all modules referred to will be left R-modules. We let g ld(R) denote the left global dimension of R, and, if M is a module, pd(M) the projective dimension of M (see e.g. [8]). A continuous chain of modules is a sequence {M~lv<c~} such that for all v, M~ is a submodule of My+l, and for all limit ordinals a, M~= U M~. Auslan- der proved the following result which he used as a lemma in proving the Global Dimension Theorem. Auslander's Lemma ([1]). Let {Mv[v<~} be a continuous chain of modules such that pd(Mo) < n and for all v + 1 < c~, pd(M~+ l/My) < n. Then pd(~) M~) < n. In this paper we shall prove a converse to a strong version of Auslander's Lemma, thereby giving a lower bound to pd( [.J M~) in terms of the projective v<cc dimension of quotients Mu/M~ (v</~<~) of modules in the chain (Theo- reml.6). The converse says, roughly, that pd(M)>n provided "sufficiently many" of the quotients MJM~ have projective dimension >n; the precise meaning of "sufficiently many" is given in terms of the notion of a stationary set. (Set-theoretic terminology is defined in Sect. 0.) For this converse we need to impose some additional assumptions, namely, that c~ is a regular uncountable cardinal (e.g. c~=Ng+ 1 for any k); that every M~ is generated by fewer than e elements; and, in addition, "coherence" assump- tions. For example, here is a very useful - though not the most general - version of the main result. Main Theorem (version III). Let R be a left coherent ring, and let M be an R- module such that every finitely generated submodule of M is finitely presented. Suppose M is the union of a continuous chain {M~JV<Nk+1} of R-modules such * Partially supported by National ScienceFoundation Grant MCS-8003591 0025-5874/82/0180/0001/$01.80