Riemann-Roch for homotopy invariant K-theory and Gysin morphisms

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We prove the Riemann-Roch theorem for homotopy invariant K-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism. In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant K-theory, motivic cohomology, their arithmetic counterparts, real absolute Hodge and Deligne-Beilinson cohomology, rigid syntomic cohomology, mixed Weil cohomologies) and use this construction to prove a motivic version of the Riemann-Roch.