Proposition 6.4.11 ([Lurie (2017), Proposition 7.2.2.6]). Let \(C\) be a stable \(\infty \)-category equipped with a t-structure, and let \(P \in C_{\geq 0}\) be a connective object. Then the following conditions are equivalent:
- (1)
- (2)
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For every \(Q \in C_{\geq 0}\) and every \(i > 0\), the abelian group \(\Ext ^i_C(P,Q)\) is zero;
- (3)
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For every \(Q \in C_{\geq 0}\), the abelian group \(\Ext ^1_C(P,Q)\) is zero;
- (4)
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Given an exact sequence \[ N' \to N \to N'' \] in \(C\) with \(N',N,N'' \in C_{\geq 0}\), the map \(\Ext ^0_C(P,N) \to \Ext ^0_C(P,N'')\) is surjective.
If \(C\) is left complete, this is further equivalent to:
Proof. Recall from Observation 6.4.3 that \(P\) is t-projective if and only if the mapping spectrum functor \(\hom _C(P,-)\colon C \to \Sp \) is right t-exact. By definition, this means that for any object \(Q \in C_{\geq 0}\), the mapping spectrum \(\hom _C(P,Q)\) has vanishing negative homotopy groups \(\pi _{-i}\hom _C(P,Q) = \Ext ^i_C(P,Q)\) for \(i > 0\). This shows the equivalence \(\text{(1)} \Leftrightarrow \text{(2)}\).
It is clear that (2) implies (3), and the implication \(\text{(3)} \Rightarrow \text{(2)}\) follows by replacing \(Q\) by \(Q[i-1]\) for all \(i \geq 1\).
We next show that \(\text{(3)} \Leftrightarrow \text{(4)}\). It is clear that (3) implies (4): the exact sequence \(N' \to N \to N''\) induces a long exact sequence of the form \[ \dots \to \Ext ^0_C(P,N) \to \Ext ^0_C(P,N'') \to \Ext ^1_C(P,N') \to \dots , \] so the vanishing of \(\Ext ^1_C(P,N')\) implies that the first map must be surjective. Assume now that (4) holds. Given a class \(\eta \in \Ext ^1(P,Q) = \pi _0\Hom _C(P,Q[1])\), we obtain an exact sequence \[ P' \to P \xrightarrow {\eta } Q[1] \] and thus a long exact sequence \[ \dots \to \Ext ^0_C(P,P') \to \Ext ^0_C(P,P) \xrightarrow {\eta \circ -} \Ext ^1_C(P,Q) \to \dots \] As \(P,P',Q \in C_{\geq 0}\), the first map is surjective, hence the second map is the zero map. Since it sends \(\id _P\) to \(\eta \), we conclude that \(\eta = 0\). This shows that \(\text{(3)} \Leftrightarrow \text{(4)}\).
Finally, suppose that \(C\) is left complete. It is clear that (2) implies (5). Conversely, assume (5). The exact sequences \[ (\pi _{n+1}Q)[n+1] \to \tau _{\leq n+1}Q \to \tau _{\leq n}Q \] show that the towers of Ext-groups in each positive degree are eventually constant. Since left completeness gives \(Q \simeq \lim _n \tau _{\leq n}Q\), the corresponding tower of mapping spectra computes \(\hom _C(P,Q)\). Eventual constancy in adjacent degrees eliminates the derived-limit term and gives \[ \Ext ^i_C(P,Q) \cong \Ext ^i_C(P,\tau _{\leq 0}Q)=0 \] for \(i>0\). This is the Postnikov-tower argument of [Lurie (2017), Proposition 7.2.2.6]. □
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