2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142744It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions.5 pages formatted for AGMF proceedingsRings and AlgebrasK-Theory and Homology18G25text