In classical algebra, flat modules form another important class of ‘well-behaved’ modules. An \(R\)-module \(N\) is called flat if the tensor product functor \(- \otimes _R N\) is exact: short exact sequences of \(R\)-modules remain exact after tensoring with \(N\). While projective modules are always flat, the converse does not hold in general. For example, \(\Q \) is a flat \(\Z \)-module but not projective.
Just as the notion of projective objects generalizes to t-projective objects in a stable \(\infty \)-category with t-structure, the notion of flat objects generalizes to the following setting.
Definition 6.5.1. Let \(C\), \(D\), and \(E\) be stable \(\infty \)-categories equipped with t-structures. Let \(- \otimes - \colon C \times D \to E\) be a functor that is exact in each variable separately, and assume it restricts to \(C_{\geq 0} \times D_{\geq 0} \to E_{\geq 0}\), i.e. the tensor product of connective objects is connective.
A connective object \(N \in D_{\geq 0}\) is called t-flat (with respect to \(\otimes \)) if the functor \((-) \otimes N \colon C \to E\) is t-exact. Similarly, \(M \in C_{\geq 0}\) is t-flat if the functor \(M \otimes (-) \colon D \to E\) is t-exact.
Remark 6.5.2. By assumption, \(- \otimes N\) is right t-exact, so \(N\) is t-flat if and only if it is additionally left t-exact.
Remark 6.5.3 (Motivation). The three main examples we have in mind for this abstract setup are:
- (1)
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The tensor product \(\Mod _R \times \Mod _R \xrightarrow {\otimes _R} \Mod _R\) for a commutative ring spectrum \(R\).
- (2)
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More generally the relative tensor product \(\RMod _R \times \LMod _R \xrightarrow {\otimes _R} \Sp \) for an associative ring spectrum \(R\).
- (3)
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The derived tensor product \(\D ^-(\Aa ) \times \D ^-(\Aa ) \xrightarrow {\otimes ^{\bL }} \D ^-(\Aa )\) for a symmetric monoidal abelian category \(\Aa \) with enough projectives (constructed in Section 6.6).
We work in this generality for two reasons. First, it makes clear that the notion of flatness does not depend on the intricate coherences involved in the definition of \(\Mod _R\) and its tensor product, discussed in Part II of this book. Second, it provides a single formulation that covers all relevant examples.
Exercise 6.5.4. Show that the t-flat objects in \(C\) and \(D\) are closed under finite direct sums and retracts.
The following exercise highlights the structural analogy between t-projectivity and t-flatness.
Exercise 6.5.5. Let \(C\) be a stable \(\infty \)-category with a t-structure. Setting \(D := C\catop \) and \(E := \Sp \catop \), the hom spectrum functor in \(C\) takes the form \(\hom _C\colon C \times C\catop \to \Sp \catop \).
- (1)
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Show that this functor satisfies the conditions of Definition 6.5.1: For \(X \in C_{\geq 0}\) and \(Y \in C_{\leq 0}\) we have \(\hom _C(X,Y) \in \Sp _{\leq 0}\).
- (2)
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Show that an object \(P \in C\) is t-flat with respect to \(\hom _C\) if and only if it is t-projective.
Just as the homotopy groups of mapping spectra give rise to Ext-groups, the homotopy groups of tensor products give rise to Tor-groups.
Definition 6.5.6 (Tor-groups). In the situation of Definition 6.5.1, for objects \(M \in C\) and \(N \in D\), we define the \(n\)-th Tor-group to be \[ \Tor ^{\otimes }_n(M,N) \quad := \quad \pi _n(M \otimes N) \qin E^{\heartsuit }. \] When \(M \in C^{\heartsuit }\) and \(N \in D^{\heartsuit }\) both lie in the heart, we also define \(M \otimes ^{\heartsuit } N := \Tor ^{\otimes }_0(M,N) = \pi _0(M \otimes N)\), resulting in a functor \[ - \otimes ^{\heartsuit } - \colon C^{\heartsuit } \times D^{\heartsuit } \to E^{\heartsuit }. \] Since \(M \otimes N \in E_{\geq 0}\), this agrees with \(\tau _{\leq 0}(M \otimes N)\) and it follows that this functor is right exact in both variables.
Observation 6.5.7 (Tor long exact sequence). Every exact sequence \(N' \to N \to N''\) in \(D\) induces an exact sequence \(M \otimes N' \to M \otimes N \to M \otimes N''\) in \(E\) (since \(M \otimes (-)\) is exact), and thus a long exact sequence of homotopy groups of the form \[ \begin {aligned} \dots \to \Tor ^{\otimes }_{n+1}(M,N'') \to \Tor ^{\otimes }_{n}(M,N') &\to \Tor ^{\otimes }_{n}(M,N) \\ &\to \Tor ^{\otimes }_{n}(M,N'') \to \Tor ^{\otimes }_{n-1}(M,N') \to \dots . \end {aligned} \] As a special case, given an object \(N \in D^{\heartsuit }\) and a short exact sequence \(M' \hookrightarrow M \twoheadrightarrow M''\) in \(C^{\heartsuit }\), the negative Tor-groups vanish, giving a one-sided long exact sequence of the form \[ \begin {aligned} \dots \to \Tor ^{\otimes }_1(M,N) \to \Tor ^{\otimes }_1(M'',N) &\to \Tor ^{\otimes }_{0}(M',N) \\ &\to \Tor ^{\otimes }_{0}(M,N) \to \Tor ^{\otimes }_{0}(M'',N) \to 0. \end {aligned} \] The same remarks apply to the other variable.
Exercise 6.5.8. The derived \(\infty \)-category \(\D (\Z )\) carries a symmetric monoidal structure \(\otimes ^{\bL }\) with unit \(\Z \); see Section 6.6. For an abelian group \(M\), use the short exact sequence \[ 0 \to \Z \xrightarrow {\cdot n} \Z \to \Z /n\Z \to 0 \] to show that \(\Tor ^{\otimes ^{\bL }}_1(M, \Z /n\Z )\) is the subgroup of \(M\) consisting of the \(n\)-torsion elements, i.e., those \(x \in M\) satisfying \(nx = 0\).
We now give several equivalent characterizations of t-flatness, analogous to Proposition 6.4.11 for t-projective objects.
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. □
In practice, we may often reduce to the case \(i = 1\) in condition (4). This is an instance of the following lemma:
Lemma 6.5.10. Assume that \(C\) is right complete, that \(E_{\leq 0}\) is closed under sequential colimits, that \((-)\otimes N\) preserves sequential colimits, and that every object \(M \in C^{\heartsuit }\) admits an epimorphism \(P \twoheadrightarrow M\) from a t-flat object \(P\). Then an object \(N \in D^{\heartsuit }\) is t-flat if and only if \(\Tor ^{\otimes }_1(M,N) = 0\) for every \(M \in C^{\heartsuit }\).
Proof. The ‘only if’ is clear from Proposition 6.5.9. Conversely, we show by induction on \(i\) that \(\Tor _i^{\otimes }(M,N) = 0\) for all \(M \in C^{\heartsuit }\) and all \(i > 0\). The case \(i = 1\) holds by assumption. Assuming the claim in degree \(i-1\), consider an epimorphism \(P \twoheadrightarrow M\) with \(P\) t-flat, and let \(M'\) be its kernel. Then the long exact sequence of Tor-groups contains \[ \dots \to \Tor _i^{\otimes }(P,N) \to \Tor _i^{\otimes }(M,N) \to \Tor _{i-1}^{\otimes }(M',N) \to \Tor _{i-1}^{\otimes }(P,N) \to \dots . \] The first term vanishes because \(P\) is t-flat, and the third vanishes by the induction hypothesis applied to \(M'\). Hence \(\Tor _i^{\otimes }(M,N)=0\), and Proposition 6.5.9 shows that \(N\) is t-flat. □
As a consequence of Proposition 6.5.9, we may compute Tor-groups in terms of flat resolutions.
Definition 6.5.11. Let \(N \in D^{\heartsuit }\) be an object in the heart of \(D\). A flat resolution of \(N\) is a long exact sequence \[ \dots \to F_2 \to F_1 \to F_0 \twoheadrightarrow N \to 0 \] in \(D^{\heartsuit }\) such that each object \(F_n\) is t-flat.
Proposition 6.5.12 (Tor-groups via flat resolutions). Let \(F_{\bullet }\) be a flat resolution of \(N \in D^{\heartsuit }\). Then for every object \(M \in C^{\heartsuit }\) and every \(n \in \Z \) there is an isomorphism \[ \Tor ^{\otimes }_n(M,N) \quad \cong \quad H_n(M \otimes ^{\heartsuit } F_{\bullet }) \] between the \(n\)-th Tor-group and the \(n\)-th homology group of the chain complex \(M \otimes ^{\heartsuit } F_{\bullet }\) in \(E^{\heartsuit }\).
Proof. The proof is entirely dual to Proposition 6.4.14; we will therefore be rather brief. Set \(N_{-1}:=N\). The flat resolution splits up into short exact sequences of the form \[ 0 \to N_n \rightarrowtail F_n \twoheadrightarrow N_{n-1} \to 0. \] This provides a long exact sequence on Tor-groups by Observation 6.5.7, in which the higher Tor-groups of \(F_n\) vanish by Proposition 6.5.9. Using that \(M \otimes ^{\heartsuit } -\colon D^{\heartsuit } \to E^{\heartsuit }\) is right exact, one obtains isomorphisms \[ M \otimes ^{\heartsuit } N_{k-1} \quad \cong \quad \coker (M \otimes ^{\heartsuit } F_{k+1} \to M \otimes ^{\heartsuit } F_k) \] for all \(k \geq 0\), which in turn give isomorphisms \[ \Tor ^{\otimes }_1(M,N_{k-1}) \quad \cong \quad \frac {\ker (M \otimes ^{\heartsuit } F_{k+1} \to M \otimes ^{\heartsuit } F_{k})}{\im (M \otimes ^{\heartsuit } F_{k+2} \to M \otimes ^{\heartsuit } F_{k+1})} \quad = \quad H_{k+1}(M \otimes ^{\heartsuit } F_{\bullet }) \] for all \(k \geq 0\). The preceding cokernel formula for \(k=0\) gives the claim in degree zero. Finally, one can inductively show that \(\Tor ^{\otimes }_n(M,N) \cong \Tor ^{\otimes }_1(M,N_{n-2})\) for all \(n \geq 1\). Setting \(k = n-1\) in the previous isomorphism thus gives \[ \Tor ^{\otimes }_n(M,N) \quad \cong \quad H_{n}(M \otimes ^{\heartsuit } F_{\bullet }) \] for all \(n \geq 1\), finishing the proof. □
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