where and are lattices in , and for some and , respectively. It is a morphism of complexes, so it induces a cup product in Deligne cohomology.
For , the Beilinson-Deligne cup product is associative and commutative up to homotopy, so it induces an associative and commutatvive cup product on differential cohomology
Pierre Deligne, Théorie de Hodge II , IHES Pub. Math. (1971), no. 40, 5–57.
Alexander Beilinson, Higher regulators and values of L-functions , J. Soviet Math. 30 (1985), 2036—2070
Alexander Beilinson, Notes on absolute Hodge cohomology , Applications of algebraic -theory to algebraic geometry and number theory, Part I, II, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.
Helene Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)