The Stacks project

Lemma 22.27.5. Let $(A, \text{d})$ be a differential graded algebra. For every compact object $E$ of $D(A, \text{d})$ there exist integers $a \leq b$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(E, M) = 0$ if $H^ i(M) = 0$ for $i \in [a, b]$.

Proof. Observe that the collection of objects of $D(A, \text{d})$ for which such a pair of integers exists is a saturated, strictly full triangulated subcategory of $D(A, \text{d})$. Thus by Proposition 22.27.4 it suffices to prove this when $E$ is represented by a differential graded module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. Using the compatibility with triangles, we see that it suffices to prove it for $P = A$. In this case $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(A, M) = H^0(M)$ and the result holds with $a = b = 0$. $\square$


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