Derived p-completion can have an L1-term

Warning 7.3.13. The group \(L_1^p A\) detects compatible systems of \(p\)-power torsion, and it need not vanish. For example, let \(X := H(\Z /p^{\infty })\). Since the Prüfer group is injective, \(L_0^p(\Z /p^{\infty })=0\), whereas Lemma 7.3.9 gives \(L_1^p(\Z /p^{\infty }) \cong \Z _p\). Hence \[ X^{\wedge }_p \; \simeq \; H(\Z _p)[1]. \] This example in particular shows that completion is not computed degreewise on homotopy groups: the \(p\)-completion of a nonzero spectrum concentrated in degree \(0\) may be concentrated in degree \(1\).