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

Hirzebruch-Riemann-Roch and Lefschetz type formulas for finite dimensional algebras