nLab
twisted module of homomorphisms

Let CC be a dg-coalgebra, AA a dg-algebra, NN a left CC-dg-comodule with coaction δ N:NCN\delta_N:N\to C\otimes N, PP a left AA-dg-module with action m P:APPm_P:A\otimes P\to P and τ:CA\tau:C\to A a twisting cochain. The twisted module of homomorphisms Hom τ(N,P)\mathrm{Hom}_\tau(N,P) is a chain complex which as a graded module coincides with the ordinary module of homomorphisms of the underlying chain complex Hom(N,P)\mathrm{Hom}(N,P), and with the differential d τd_\tau given by

d τ(f)=d(f)+m P(τf)δ N, d_\tau(f) = d(f) + m_P\circ(\tau\otimes f)\circ\delta_N,

where fHom(N,P)f\in\mathrm{Hom}(N,P).

Revised on March 15, 2009 22:32:33 by Zoran Škoda (195.37.209.180)