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 $[D,p] = [im(X_{\omega}: D_{\omega}\to D_{\omega})] - [D_{odd}]\in K_ 0(A).$

MSC:

 57Q12 Wall finiteness obstruction for CW-complexes 55U10 Simplicial sets and complexes in algebraic topology