Do etale $K$-theory of finite fields coincide with algebraic $K$-theory? what is the easiest way to see this (I think etale and algebraic $K$-theory coincide above a certain value like cohomological dimension).
According to this theorem 4.4.3. The Bott inverted $K$-theory of finite field is given which should coincide with etale $K$-theory. Now another question, why the Bott inverted $K$-theory given above is not 2-periodic? Isn't it supposed that all even degrees should match and similarly for the odd degrees?