Corollary 6.6.28. Assume that all projective objects in \(\Aa \) are flat.
- (1)
-
The derived tensor product may be computed by resolving only one variable: for any projective resolutions \(P\to C\) and \(Q\to D\), there are isomorphisms \[ C \otimes ^{\bL } D \simeq P\otimes D \simeq C\otimes Q \] in \(\D ^-(\Aa )\).
- (2)
-
Tor groups in \(\Aa \) may be computed by resolving one variable: given objects \(M, N \in \Aa \), we have \[ \Tor _n^{\Aa }(M, N) \; \cong \; H_n(P_{\bullet } \otimes N) \] for any projective resolution \(P_{\bullet } \to M\).
- (3)
-
For a flat object \(F \in \Aa \), the complex \(F[0]\) is a t-flat object of \(\D ^-(\Aa )\).
Proof. Part (1) is immediate from Lemma 6.6.27, as \(P\) and \(Q\) are bounded below flat chain complexes. Part (2) then follows, as \(\Tor _n^{\Aa }(M, N)\) is defined as the \(n\)-th homology group of \(M \otimes ^{\bL } N\). Part (3) also follows, as the derived tensor \(F \otimes ^{\bL } -\) is quasi-isomorphic to the underived tensor \(F \otimes -\), which preserves both connective and coconnective complexes. □
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