The relative dihedral homology of involutive algebras (Q1970285)

Given \(A\), \(B\) involutive algebras and \(f\colon A\to B\) a homomorphism, the author defines a free involutive algebra resolution \(R\) of the algebra \(B\) over \(f=\pi i\), \(i\colon A\to R\) the inclusion, \(\pi\colon R\to B\) a quasiisomorphism and then considers the relative dihedral homology as: \[ {^\varepsilon HD_*}(A@>f>>B)=H_*\bigl(\tfrac R{A+[R,R]+\text{Im}(1-r^\varepsilon)}\bigr), \] where \([R,R]\) is the commutant of the algebra \(R\), \(r^\varepsilon\) is the involution on \(R\). Then he studies its main properties: (i) For \(A\) an involutive algebra \({^\varepsilon H}D_i(A\to 0)={^\varepsilon H}D_{i-1}(A)\). (ii) \({^\varepsilon H}D_i(A@>f>>B)\) does not depend on the choice of the resolution. (iii) For \(A\), \(B\) and \(C\) involutive algebras the sequence \(A@>f>>B@>g>>C\) induces a long exact sequence of relative dihedral homology: \[ \cdots\to{^\varepsilon H}D_i(A@>f>>B)\to{^\varepsilon H}D_i(A@>gf>>C)\to{^\varepsilon H}D_i(B@>g>>C)\to{^\varepsilon H}D_{i-1}(A\to B)\to\cdots\;. \]