The differential – or Tamarkin-Tsygan calculus – of an associative algebra provides rich information about the algebra, it englobes Hochschild (co)homology with its algebraic structure and the Connes differential. It has been proven recently by Armenta and Keller that it is a derived invariant. On the other hand a relative version of Hochschild cohomology exists from long, it has been rarely used because of the lack of relation with the usual theory. Kaygun has obtained in 2012 a long exact sequence relying the theories. A main purpose will be to determine if a relative differential – or Tamarkin-Tsygan – calculus exists, in particular a relative Connes differential and to establish a Jacobi-Zariski-like relation with the usual calculus.
​Our second aim is to show that the class of algebras satisfying Han’s conjecture is closed under split extensions of algebras by any multiplicative bimodule which is just of finite projective dimension and tensor nilpotent.