Riemann–Roch like theorem for triangulated categories

We will prove a Riemann–Roch like theorem for triangulated categories satisfying Serre duality. As an application, we will prove Riemann–Roch Theorem and Adjunction Formula for noncommutative Cohen–Macaulay surfaces in terms of sheaf cohomology. We will also show that the formulas hold for stable categories over Koszul connected graded algebras in terms of Tate–Vogel cohomology by extending the BGG correspondence.