An obstruction to lifting cyclic modules

Melvin Hochster

Vol. 61 (1975), No. 2, 457–463

DOI: 10.2140/pjm.1975.61.457

Abstract

Let A be a ring (all ring are commutative, with identity), let t ∈ A be a nonzerodivisor and not a unit, and let B = A∕At. Let M be a B-module of finite type. We call an A-module E of finite type a lifting of M (or we say that “E lifts M” or “M lifts to E”) if (1) t is not a zerodivisor on E and (2) E∕tE≅M. Grothendieck’s lifting problem (GLP) is this: Suppose that (A,m) is a complete regular local ring and that t ∈ m − m², so that B = A∕tA is again regular. If M is a B-module of finite type, does M lift to A? A simple and completely elementary counterexample is given below for the case where M is cyclic.

Mathematical Subject Classification 2000

Primary: 13C10

Milestones

Received: 13 May 1975

Published: 1 December 1975