A note on cyclotomic Euler systems and the double complex method

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In this note we give the Kolyvagin recursion in cyclotomic Euler systens a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of the chosen field satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution a la Kubert. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.