First derived limit

Definition 4.4.36 (First derived limit). Let \(\dots \xrightarrow {f_2} A_2 \xrightarrow {f_1} A_1 \xrightarrow {f_0} A_0\) be a tower of abelian groups. Define a homomorphism \[ d\colon \prod _{k \geq 0}A_k \longrightarrow \prod _{k \geq 0}A_k, \qquad d(a)_k := a_k-f_k(a_{k+1}). \] The first derived limit of the tower is the abelian group \[ \lim _k^1 A_k := \coker (d). \] Notice that the ordinary inverse limit \(\lim _k A_k\) is the kernel of \(d\).