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For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor $$ W_r : \mathrm{wEt}_A \longrightarrow \mathrm{wEt}_{W_r(A)} $$ is well-defined and commutes with all colimits.

Can one explicit a right adjoint to $W_r$ ?

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    $\begingroup$ Greenberg functor? $\endgroup$
    – Jason Starr
    Commented Jul 20, 2016 at 14:52
  • $\begingroup$ Can you give a reference, or expand your answer ? $\endgroup$
    – js21
    Commented Aug 17, 2016 at 8:44

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