= Surjectivity on adic completions
{title2=$M\twoheadrightarrow N\implies\widehat M\twoheadrightarrow\widehat N$}
For any ideal $I$ and surjective module map $f:M\to N$, compatible representatives $n_r$ modulo $I^rN$ can be lifted recursively. Having chosen $f(m_r)=n_r$, lift $n_{r+1}-n_r\in I^rN$ to $\delta_r\in I^rM$ and put $m_{r+1}=m_r+\delta_r$. These define the required element of the <adic completion of a module>. No finite-generation hypothesis on the modules is required.
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