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A survey of obstruction theory for projective modules of top rank. (English) Zbl 0962.13008
Lam, T. Y. (ed.) et al., Algebra, $$K$$-theory, groups, and education. On the occasion of Hyman Bass’s 65th birthday. Mainly the proceedings of the conference, Columbia University, New York, NY, November 6-7, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 243, 153-174 (1999).
The paper is a survey on results concerning obstruction to split off a free module of rank one from a projective module of rank equal to $$\dim A$$ over a noetherian ring $$A$$. First there are given the stability theorems of Bass and Serre which were the starting point for the study of structure of projective modules over arbitrary noetherian rings, as well as algebraic $$K$$-theory. Results of Suslin and Mohan Kumar are stated about cancellation of projective modules. Also, the author surveys briefly results about indecomposable projective modules of rank equal to the dimension of the ring and results concerning the top Chern class as an obstruction to splitting off a free summand of rank one from projective modules of rank equal to the dimension of the ring, when the ring in question is the coordinate ring of an affine variety over an algebraically closed field. A major part of this paper surveys recent work of S. M. Bhatwadekar and R. Sridharan [see e.g. Invent. Math. 136, No. 2, 287-322 (1999; Zbl 0949.14005)] concerning the notion of Euler class groups of rings and Euler classes of projective modules of top rank, introduced by M. V. Nori.
For the entire collection see [Zbl 0928.00073].

##### MSC:
 13C10 Projective and free modules and ideals in commutative rings