A nontrivial Postnikov extension

Example 6.4.15 (A nontrivial Postnikov extension). Let \(C\) be a stable \(\infty \)-category with a t-structure, and let \(X\in C_{\geq 0}\cap C_{\leq 1}\). Its truncation sequence takes the form \[ \pi _1(X)[1]\longrightarrow X\longrightarrow \pi _0(X) \xrightarrow {\ \kappa _X\ }\pi _1(X)[2]. \] We call the final morphism \[ \kappa _X\in \Ext ^2_C(\pi _0(X),\pi _1(X)) \] the first Postnikov \(k\)-invariant of \(X\). The exact sequence recovers \(X\) as the fiber of \(\kappa _X\). In particular, if \(\kappa _X\) is zero, then \(X\cong \pi _0(X)\oplus \pi _1(X)[1]\). The \(k\)-invariant therefore records how the two homotopy group objects are glued together.

For a concrete nontrivial example, set \(R:=\Z /4\) and \(K:=R/(2)\cong \Z /2\), and consider the complex \[ A_{\bullet }:=\big (0\longrightarrow R\xrightarrow {\ 2\ }R\longrightarrow 0\big ) \] in \(\D (R)\), with the two copies of \(R\) in degrees \(1\) and \(0\). Its only nonzero homology groups are \[ H_1(A_{\bullet })\cong K \qquad \text {and}\qquad H_0(A_{\bullet })\cong K. \] The periodic projective resolution \[ \cdots \xrightarrow {\ 2\ }R\xrightarrow {\ 2\ }R \xrightarrow {\ 2\ }R\twoheadrightarrow K\longrightarrow 0 \] and Proposition 6.4.14 show that \(\Ext ^n_R(K,K)\cong K\) for every \(n\geq 0\): after applying \(\Hom _R(-,K)\), all differentials vanish. In the Yoneda description from Exercise 6.4.9, the first \(k\)-invariant of \(A_{\bullet }\) is represented by the exact \(2\)-extension \[ 0\longrightarrow K\longrightarrow R\xrightarrow {\ 2\ }R \longrightarrow K\longrightarrow 0, \] which is obtained from the first two steps of this resolution and hence represents the nonzero element of \(\Ext ^2_R(K,K)\). Consequently, \[ A_{\bullet }\not \cong K[1]\oplus K[0] \] in \(\D (R)\), although the two sides have isomorphic homology groups in every degree.