The Stacks project

Lemma 15.76.5. Let $R$ be a ring. Let $K \in D^-(R)$. Let $a \in \mathbf{Z}$. Assume that for any injective $R$-module map $M \to M'$ the map $\mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(K, M) \to \mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(K, M')$ is injective. Then there is a unique direct sum decomposition $K \cong \tau _{\leq a}K \oplus \tau _{\geq a + 1}K$ and $\tau _{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$.

Proof. Consider the distinguished triangle

\[ \tau _{\leq a}K \to K \to \tau _{\geq a + 1}K \to (\tau _{\leq a}K)[1] \]

in $D(R)$, see Derived Categories, Remark 13.12.4. Observe that $\mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(\tau _{\leq a}K, M) = \mathop{\mathrm{Hom}}\nolimits _ R(H^ a(K), M)$ and $\mathop{\mathrm{Ext}}\nolimits ^{-a - 1}_ R(\tau _{\leq a}K, M) = 0$, see Derived Categories, Lemma 13.27.3. Thus the long exact sequence of $\mathop{\mathrm{Ext}}\nolimits $ gives an exact sequence

\[ 0 \to \mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(\tau _{\geq a + 1}K, M) \to \mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(K, M) \to \mathop{\mathrm{Hom}}\nolimits _ R(H^ a(K), M) \]

functorial in the $R$-module $M$. Now if $I$ is an injective $R$-module, then $\mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(\tau _{\geq a + 1}K, I) = 0$ for example by Derived Categories, Lemma 13.27.2. Since every module injects into an injective module, we conclude that $\mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(\tau _{\geq a + 1}K, M) = 0$ for every $R$-module $M$. By Lemma 15.68.2 we conclude that $\tau _{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$ (this is where we use that $K$ is bounded above). We obtain the splitting by Lemma 15.76.1. $\square$

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