Lück, Wolfgang; Ranicki, Andrew Chain homotopy projections. (English) Zbl 0616.57011 Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 73, 31 p. (1986). Our main result is a generalization of the instant finiteness obstruction to chain homotopy projections: Let A be an associative ring with 1. A near-projection of an A-module M is an endomorphism \(p: M\to M\) such that \((p(1-p))^ N=0: M\to M\) for some integer \(N\geq 0\), in which case standard algebra leads to a projection \[ p_{\omega}=(p^ N+(1-p)^ N)^{-1}p^ N: M\to M\quad. \] Theorem. Given a chain homotopy projection (D,p) with D a finite chain complex of f.g. projective A- modules and a chain homotopy \(q:p^ 2\simeq p: D\to D\) there is defined a near-projection \[ \begin{split} X= \begin{pmatrix} p& -d& 0& ...\\ -q& 1-p& d& ... \\ 0& q& p& ... \\ \vdots & \vdots & \vdots \end{pmatrix} :\\ D_{\omega} = D_ 0\oplus D_ 1\oplus D_ 2\oplus... \to D_{\omega} = D_ 0\oplus D_ 1\oplus D_ 2\oplus... \end{split} \] such that \[ [D,p] = [im(X_{\omega}: D_{\omega}\to D_{\omega})] - [D_{odd}]\in K_ 0(A). \] Cited in 1 Document MSC: 57Q12 Wall finiteness obstruction for CW-complexes 55U10 Simplicial sets and complexes in algebraic topology Keywords:instant finiteness obstruction; chain homotopy projections; near- projection × Cite Format Result Cite Review PDF