Comparing direct limit and inverse limit of even <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -groups in multiple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml

Iwasawa first established a duality relating the direct limit and the inverse limit of class groups in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -extension, and this result has recently been extended to multiple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -extensions by many authors. In this paper, we establish an analogous duality for the direct limit and the inverse limit of higher even <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -groups in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mi>p</mml:mi> <mml:mi>d</mml:mi> </mml:msubsup> </mml:math> -extension. We then give some examples where the direct limit may or may not vanish.