T-spectra and Poincar\'e Duality

Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring-spectrum and proved the Poincar\'e Duality theorem for this general case. In the current paper we consider oriented cohomology theories on algebraic varieties represented by multiplicative symmetric $T$-spectra and prove the Duality theorem, which mimics the result of Adams. This result is held, in particular, for Motivic Cohomology and Algebraic Cobordism of Voevodsky.