Let \((I,\leq)\) be a directed set. An inverse system of abelian groups indexed by \(I\) consists of abelian groups \(A_\alpha\) for \(\alpha\in I\) and homomorphisms \(f_{\beta\alpha}\colon A_\beta\to A_\alpha\) for \(\alpha\leq\beta\), with \(f_{\alpha\alpha}=\id_{A_\alpha}\) and \(f_{\beta\alpha}\circ f_{\gamma\beta}=f_{\gamma\alpha}\) for \(\alpha\leq\beta\leq\gamma\). Its inverse limit is the subgroup of compatible families \[ \varprojlim_\alpha A_\alpha\coloneqq\ts{(a_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}A_\alpha \st f_{\beta\alpha}(a_\beta)=a_\alpha\text{ whenever }\alpha\leq\beta}, \] with the projections \(\pi_\alpha\colon\varprojlim A_\alpha\to A_\alpha\).
Inverse limits are left exact: if \(0\to A_\alpha\mapsvia{u_\alpha}B_\alpha\mapsvia{v_\alpha}C_\alpha\) is exact for every \(\alpha\in I\) and the \(u_\alpha\) and \(v_\alpha\) commute with the maps of the inverse systems, then the induced sequence \(0\to\varprojlim A_\alpha\mapsvia{u}\varprojlim B_\alpha\mapsvia{v}\varprojlim C_\alpha\) is exact.
For \(I=\NN\), write \(f_n\colon A_{n+1}\to A_n\) for the bonding maps and define \(\delta\colon\prod_nA_n\to\prod_nA_n\) by \(\delta((a_n)_n)\coloneqq(a_n-f_n(a_{n+1}))_n\). Then \(\ker\delta=\varprojlim A_n\), and \(\varprojlim^1A_n\coloneqq\coker\delta\). A short exact sequence \(0\to A_n\to B_n\to C_n\to0\) of inverse systems indexed by \(\NN\) gives an exact sequence \[ 0\to\varprojlim A_n\to\varprojlim B_n\to\varprojlim C_n\to\varprojlim{}^1A_n\to\varprojlim{}^1B_n\to\varprojlim{}^1C_n\to0, \] so \(\varprojlim^1\) measures the failure of \(\varprojlim B_n\to\varprojlim C_n\) to be surjective [@Hat02].