Failure of Cancellation for Direct Sums of Line Bundles

Abstract

In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules L of rank 1 over a commutative ring R suchthat L®L~X is stably free but not free.Examples are also given of projective modules for which the determinant map det: Aut(P) -» R* is not onto.Some related topological questions are also discussed.The following question was raised independently by M. P. Murthy and R. Wiegand.Question.Does there exist a commutative ring R and a finitely generated projective module L over R of rank 1 such that L® L~x is stably free but not free?I will show here that the answer is affirmative.We can even assume that L® L~l ®R is free.Theorem 1.There exist rings R of the following types with a finitely generated projective module L over R of rank 1 such that L®L~l®R is free but L®L~l is not free.( 1 ) R is a sub R-algebra of R[x, y], finitely generated over R, and with quotient field R(x, y).(2) R is a smooth rational domain of dimension 3, finitely generated over R.(3) R is a rational domain of dimension 5, finitely generated over C. (4) R is a smooth rational domain of dimension 1, finitely generated over C.The examples will be constructed explicitly but the proof that L © L-1 is not free will use topological methods as in [SV].In § §7 and 8, we give examples of projective modules P such that the determinant map det: Aut(P) -► R* is not onto.All topological spaces considered here will be assumed to have the homotopy type of CPT-complexes.

Locations

• Transactions of the American Mathematical Society - View - PDF