Lemma 7.3.9. Let \(A\) be an abelian group, and write \(A[p^k] \subseteq A\) for the subgroup of elements annihilated by \(p^k\). Then \[ \begin {gathered} L_1^p A \cong \lim _k A[p^k], \\[2pt] 0 \longrightarrow \lim _k^1 A[p^k] \longrightarrow L_0^p A \longrightarrow \lim _k A/p^k A \longrightarrow 0, \end {gathered} \] where the transition maps on \(A[p^k]\) are multiplication by \(p\). If \(A\) is finitely generated, then \(L_1^p A = 0\) and \(L_0^p A \cong A \otimes _{\Z } \Z _p\).
Proof. Write \(\Z /p^{\infty } \cong \colim _k \Z /p^k\) for the filtered colimit along the multiplication-by-\(p\) injections \(\Z /p^k \xrightarrow {\ \cdot p\ } \Z /p^{k+1}\). Since filtered colimits are exact in \(\Ab \), the colimit in \(\D (\Z )\) of a filtered diagram of abelian groups concentrated in degree zero is again concentrated in degree zero, with degree-zero homology given by the colimit in \(\Ab \). Thus the displayed colimit also computes the colimit in \(\D (\Z )\). Mapping out of it gives \[ \hom _{\D (\Z )}(\Z /p^{\infty },A) \simeq \lim _k \hom _{\D (\Z )}(\Z /p^k,A). \] For every \(k \geq 1\), the standard resolution of \(\Z /p^k\) identifies \[ \Hom _{\Z }(\Z /p^k,A) \cong A[p^k] \qquad \text {and}\qquad \Ext ^1_{\Z }(\Z /p^k,A) \cong A/p^k A. \] Under these identifications, the transition maps are multiplication by \(p\) on the first tower and the quotient maps on the second. The two claims now follow by applying the Milnor sequence of Corollary 4.4.37 in degrees \(0\) and \(-1\).
If \(A\) is finitely generated, its \(p\)-primary torsion is bounded, so the transition maps into every fixed stage of the first tower are eventually zero. It follows directly from the description in Definition 4.4.36 that both its limit and its first derived limit vanish. Finally, \(\lim _k A/p^k A \cong A \otimes _{\Z } \Z _p\) for every finitely generated abelian group \(A\). □
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