Proposition 6.5.9. For a connective object \(N \in D_{\geq 0}\), the following conditions are equivalent:
- (1)
- (2)
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For every \(M \in C_{\leq 0}\) and every \(i > 0\), the object \(\Tor ^{\otimes }_i(M,N)\) is zero;
- (3)
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For every \(M \in C_{\leq 0}\), the object \(\Tor ^{\otimes }_1(M,N)\) is zero;
Assume moreover that \(C\) is right complete, that the functor \((-) \otimes N\) preserves sequential colimits, and that \(E_{\leq 0}\) is closed under sequential colimits. Then these conditions are further equivalent to:
Proof. The functor \((-)\otimes N\) is right t-exact by assumption, so \(N\) is t-flat if and only if it sends \(C_{\leq 0}\) into \(E_{\leq 0}\). This is precisely condition (2). Conditions (2) and (3) are equivalent because, for \(i>0\), \[ \Tor _i^{\otimes }(M,N) \cong \Tor _1^{\otimes }(M[1-i],N), \] and \(M[1-i]\) is still coconnective.
It remains to compare with condition (4). For \(M\in C_{\leq 0}\), each truncation \(\tau _{\geq -n}M\) is obtained by finitely many extensions from shifts \((\pi _{-k}M)[-k]\) of objects of the heart. Condition (4) therefore implies that \((\tau _{\geq -n}M)\otimes N\) is coconnective. Right completeness and preservation of sequential colimits identify \(M\otimes N\) with the colimit of these objects, which is coconnective because \(E_{\leq 0}\) is closed under sequential colimits. Thus (4) implies (2); the converse is immediate. □
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