  The Hirzebuch-Riemann-Roch (HRR) and Lefschetz type formulas for finite
dimensional elementary algebras of finite global dimension are explicitly
given. They have cohomological, homological, Hochschild cohomological and
Hochschild homological four versions, and module, bimodule, module complex and
bimodule complex four levels. For this, the dimension matrix of a bimodule
(complex) and the trace matrix of a bimodule (complex) endomorphism are
introduced. It is shown that Shklyarov pairing, Chern character and
Hattori-Stallings trace can be concretely expressed by Cartan matrix, dimension
vector and trace vector in this situation. Furthermore, the HRR and Lefschetz
type formulas for finite dimensional elementary algebras of finite global
dimension and dg algebras are compared.
