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The spectral theory of the Karoubi operator due to Cuntz and Quillen is extended to general mixed (duchain) complexes, that is, chain complexes which are also cochain complexes. Connes' coboundary map B can be viewed as a perturbation of the noncommutative De Rham differential by a polynomial in the Karoubi operator. The homological impact of such perturbations is expressed in terms of two short exact sequences. U.K. thanks Gabriella Böhm, Niels Kowalzig and Tomasz Maszczyk for discussions, anddoi:10.4310/hha.2018.v20.n1.a14 fatcat:ed5bbfwwtrhgpa6gh6zttgb3my