Nakayama’s lemma for half-exact functors

Abstract

We prove an analog of Nakayama's Lemma, in which the finitely generated module is replaced by a half-exact functor from modules to modules. As applications, we obtain simple proofs of Grothendieck's property of exchange for a sheaf of modules under base change, and of the criterion for flatness. 1. Nakayama's Lemma. The form of Nakayama's Lemma we shall start with is: THEOREM 1.1 [3, ?3, PROPOSITION 11]. Let R be a commutative ring, and N a finitely generated R-module such that for all maximal ideals m c R, we have N = Nm (equivalently, N 0 (R/m) = {O}). Then N= {O}. PROOF. If N is a nonzero finitely generated R-module, we can find by Zorn's Lemma a maximal proper submodule No. Then N' = N/NO is a simple module (has no proper nonzero submodules). Every simple Rmodule is isomorphic to one of the form R/m, for m some maximal ideal. Writing N' in this form, we see that N'm = {O} $ N', so Nm $ N. REMARKS. Taking R so that there is only one maximal ideal m, we get one familiar form of Nakayama's Lemma. Another says that N = {O} if N = N91(R), where 91(R) =def nm; clearly this also follows from Theorem 1.1. The proof of that theorem can be adapted to noncommutative rings if we replace maximal ideals by right primitive ideals: (2-sided) ideals which are kernels of the action of R on simple right modules, and again one can derive local and Jacobson radical forms of the lemma. Received by the editors February 12, 1971. AMS 1970 subject class*fications. Primary 13C99, 13D99, 14A05; Secondary 13E05, 16A62, 16A64, 18E10, 18G10, 18G99.