Vol. 61, No. 2, 1975

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An obstruction to lifting cyclic modules

Melvin Hochster

Vol. 61 (1975), No. 2, 457–463

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∕tEM. 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
Received: 13 May 1975
Published: 1 December 1975
Melvin Hochster
Department of Mathematics
University of Michigan
East Hall, 530 Church Street
Ann Arbor MI 48109-1043
United States