Chain homotopy projections. (English) Zbl 0671.18005

Lück, Wolfgang; Ranicki, Andrew

J. Algebra 120, No. 2, 361-391 (1989).

If \(C_*\) is a finitely generated complex of free modules over a ring R, and p an endomorphism with \(p^ 2\) chain homotopic to p then, up to homotopy, p is a projection onto a subcomplex \(D_*\) of projective modules, which defines a class x in \(K_ 0(R)\) depending only on p and the chain homotopy. The authors develop an abstract theory of the construction of x, and describe relations with earlier work by several authors, including the topological case when p is induced by a self-map of a finite CW-complex. Here complete results are only obtained when p preserves a base point and induces the identity on the fundamental group. There is a direct construction of a module with class x (due to Ranicki); and the theory simplifies in the important case when p(1-p) is nilpotent.

Reviewer: C.T.C.Wall

MSC:

18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
57Q12	Wall finiteness obstruction for CW-complexes
18G05	Projectives and injectives (category-theoretic aspects)

Keywords:

projection onto a subcomplex of projective modules; finitely generated complex of free modules; chain homotopy; finite CW-complex

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