Negative Tate twists contribute only on their Teichmüller branch and are recovered as the fiber of γ - u^{-m} (or T = u^{-m} - 1) in the universal cyclotomic complex.
Corrected version 2.3, available athttps: //www.mathi.uni-heidelberg.de/~schmidt/NSW2e/NSW2.3.pdf
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On the cohomology of negative Tate twists via cyclotomic descent
Negative Tate twists contribute only on their Teichmüller branch and are recovered as the fiber of γ - u^{-m} (or T = u^{-m} - 1) in the universal cyclotomic complex.