Characterization of flat modules

Proof. We first compare the t-structural conditions. Apply Proposition 6.5.9 to the relative tensor product \(\RMod _R \times \LMod _R \to \Sp \). The t-structure on \(\RMod _R\) is right complete by Corollary 8.2.5, relative tensor products preserve sequential colimits, and \(\Sp _{\leq 0}\) is closed under sequential colimits. Moreover, \(M \otimes _R N\) is connective when \(M\) and \(N\) are connective by Observation 8.5.9. Thus the heart criterion of Proposition 6.5.9 says precisely that \(\text{(1)} \Leftrightarrow \text{(4)}\).

\(\text{(1)} \Rightarrow \text{(2)}\): Assume that \((-) \otimes _R N\) is t-exact. By Lemma 6.3.26, it commutes with taking homotopy group objects: for every right \(R\)-module \(M\) and \(n \in \Z \), there is a natural isomorphism \(\pi _n(M) \otimes _R N \iso \pi _n(M \otimes _R N)\) in \(\Sp ^{\heartsuit } \simeq \Ab \). Furthermore, by Corollary 6.3.17 the t-exact functor \((-) \otimes _R N\) restricts to an exact functor \(\RMod _R^{\heartsuit } \to \Ab \) on hearts. Under the equivalence \(\RMod _R^{\heartsuit } \simeq \RMod _{\pi _0(R)}^{\heartsuit }\) from Lemma 8.2.9, this functor corresponds to \((-) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N)\) by Observation 8.5.9. Combining these observations gives (2).

\(\text{(2)} \Rightarrow \text{(3)}\): Taking \(M = R\) gives the displayed isomorphisms in (3). To prove that \(\pi _0(N)\) is flat, consider a short exact sequence \(0 \to M' \to M \to M'' \to 0\) of ordinary right \(\pi _0(R)\)-modules, regarded as discrete right \(R\)-modules. Tensoring the associated exact sequence with \(N\) and applying (2) gives a short exact sequence \[ 0 \to M' \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to M \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to M'' \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to 0. \] Hence \(\pi _0(N)\) is flat over \(\pi _0(R)\).

\(\text{(3)} \Rightarrow \text{(2)}\): We prove the formula for progressively more general right \(R\)-modules \(M\).

Step 1: The isomorphism holds by assumption when \(M = R\), and hence also when \(M = F[k]\) is the shift of a free right \(R\)-module \(F = \bigoplus _I R\).

Step 2: Suppose we have an exact sequence \(M' \to M \to M''\) of right \(R\)-modules such that condition (2) holds for two of the three modules; we claim it also holds for the third. Since \(\pi _0(N)\) is a flat \(\pi _0(R)\)-module by assumption, the functor \((-) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N)\) is exact on the category of \(\pi _0(R)\)-modules. Using the long exact sequence on homotopy groups for \(M' \to M \to M''\), we obtain an exact sequence \[ \dots \to \pi _n(M') \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to \pi _n(M) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to \pi _n(M'') \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to \dots \] Comparing this with the long exact sequence for \(M' \otimes _R N \to M \otimes _R N \to M'' \otimes _R N\) via the maps from (2), the five lemma gives the claim.

Step 3: Let \(M\) be connective. The right-module version of Proposition 8.2.8 writes \(M\) as the colimit of a sequence \(M(0) \to M(1) \to \dots \) in which \(M(0)\) and all the cofibers \(\cofib (M(i) \to M(i+1))\) are shifts of free modules. Steps 1 and 2 imply the formula for every \(M(i)\). It follows for \(M\) because homotopy groups and ordinary tensor products preserve filtered colimits.

Step 4: Since the t-structure on \(\RMod _R\) is right complete (Corollary 8.2.5), an arbitrary right \(R\)-module \(M\) may be written as a filtered colimit \(M \simeq \colim _{n \leq 0} \tau _{\geq n} M\), where each \(\tau _{\geq n} M\) is a shift of a connective module. By Step 3 and the preservation of filtered colimits, (2) holds for \(M\).

\(\text{(4)} \iff \text{(5)}\): Let \(A := \pi _0(R)\), regarded as a discrete ring spectrum, and let \(L := A \otimes _R N\). Every discrete right \(R\)-module is equivalently a discrete right \(A\)-module, and for such an \(M\), associativity of relative tensor products gives a natural isomorphism \[ M \otimes _A L \iso M \otimes _R N. \] If (4) holds, then taking \(M = A\) shows that \(L\) is discrete. Moreover, the displayed isomorphism shows that \(M \otimes _A L\) is discrete for every discrete right \(A\)-module \(M\). Thus \(L\) satisfies (4) as an \(A\)-module, and hence is flat by the equivalence between (1) and (4) already proved above.

Conversely, if \(L\) is discrete and flat as an \(A\)-module, then for every discrete right \(R\)-module \(M\), regarded as a discrete right \(A\)-module, the spectrum \(M \otimes _A L\) is discrete. By the displayed isomorphism, so is \(M \otimes _R N\), which is (4).

Now assume that \(R\) and \(N\) are both discrete. We show that \(\text{(4)} \Leftrightarrow \text{(6)} \Leftrightarrow \text{(7)}\).

\(\text{(4)} \Rightarrow \text{(7)}\): This is immediate, since (4) asserts that \(\Tor ^R_i(M,N) = 0\) for all \(i > 0\).

\(\text{(7)} \Rightarrow \text{(6)}\): Let \(M' \hookrightarrow M\) be an injection of discrete right \(R\)-modules, with cokernel \(M'' := M/M'\). By Corollary 6.3.16, the sequence \(M' \to M \to M''\) is exact in \(\RMod _R\). The Tor long exact sequence (Observation 8.5.10) gives \[ \Tor ^R_1(M'',N) \to M' \otimes ^{\heartsuit }_R N \to M \otimes ^{\heartsuit }_R N \to M'' \otimes ^{\heartsuit }_R N \to 0. \] By (7), we have \(\Tor ^R_1(M'',N) = 0\), so the map \(M' \otimes ^{\heartsuit }_R N \to M \otimes ^{\heartsuit }_R N\) is injective.

\(\text{(6)} \Rightarrow \text{(4)}\): We first show that \(\Tor ^R_1(M,N) = 0\) for any discrete right \(R\)-module \(M\). Choose a free \(R\)-module \(F = \bigoplus _I R\) with a surjection \(F \twoheadrightarrow M\) (for example, take \(I\) to be the underlying set of \(M\)), and let \(K := \ker (F \twoheadrightarrow M)\). The Tor long exact sequence gives \[ \Tor ^R_1(F,N) \to \Tor ^R_1(M,N) \to K \otimes ^{\heartsuit }_R N \to F \otimes ^{\heartsuit }_R N. \] Since \(F \otimes _R N \cong \bigoplus _I N\) is discrete, we have \(\Tor ^R_1(F,N) = 0\). By (6), the inclusion \(K \hookrightarrow F\) induces an injection \(K \otimes ^{\heartsuit }_R N \hookrightarrow F \otimes ^{\heartsuit }_R N\). Exactness then forces \(\Tor ^R_1(M,N) = 0\).

For higher Tor-groups, we use dimension shifting. The short exact sequence \(0 \to K \hookrightarrow F \twoheadrightarrow M \to 0\) gives, for \(i \geq 1\), \[ \Tor ^R_{i+1}(F,N) \to \Tor ^R_{i+1}(M,N) \to \Tor ^R_i(K,N) \to \Tor ^R_i(F,N). \] Since \(F\) is free, \(\Tor ^R_j(F,N) = 0\) for \(j > 0\). Thus \(\Tor ^R_{i+1}(M,N) \cong \Tor ^R_i(K,N)\) for \(i \geq 1\). By induction, \(\Tor ^R_i(M,N) = 0\) for all \(i > 0\) and all discrete \(M\). □