In Section 6.4, Section 6.5 we developed the general theory of t-projective and t-flat objects in stable \(\infty \)-categories equipped with t-structures. In this section, we specialize this theory to modules over ring spectra, obtaining analogues of projective and flat modules from ordinary algebra.
8.5.1 Projective modules
Definition 8.5.1 (Ext-groups for modules). Let \(R\) be an associative ring spectrum and let \(M, N \in \LMod _R\) be left \(R\)-modules. We define the \(n\)-th Ext-group of \(M\) and \(N\) over \(R\) to be \[ \Ext ^n_R(M,N) \quad := \quad \Ext ^n_{\LMod _R}(M,N) \quad = \quad \pi _{-n}\hom _R(M,N) \quad \simeq \quad \pi _0 \Hom _{\LMod _R}(M,N[n]). \] These are the Ext-groups from Definition 6.4.6 specialized to the stable \(\infty \)-category \(\LMod _R\).
The general vanishing, degree-zero identification, and long exact sequence for Ext-groups from Observation 6.4.7, Observation 6.4.8, Observation 6.4.10 apply verbatim to \(R\)-modules.
Definition 8.5.2 (Projective module). Let \(R\) be a connective associative ring spectrum. A left \(R\)-module \(P\) is projective if it is a t-projective object of \(\LMod _R\) in the sense of Definition 6.4.1; that is, if the mapping spectrum functor \(\hom _R(P,-)\colon \LMod _R \to \Sp \) is t-exact.
Observation 8.5.3. The characterizations of t-projective objects from Proposition 6.4.11 specialize to the following equivalent conditions for a connective left \(R\)-module \(P\):
- (1)
-
\(P\) is projective;
- (2)
-
For every connective left \(R\)-module \(Q\) and every \(i > 0\), we have \(\Ext ^i_R(P,Q) = 0\);
- (3)
-
For every connective left \(R\)-module \(Q\), we have \(\Ext ^1_R(P,Q) = 0\);
- (4)
-
Given an exact sequence \(N' \to N \to N''\) of connective left \(R\)-modules, the induced map \(\Ext ^0_R(P,N) \to \Ext ^0_R(P,N'')\) is surjective.
Since the t-structure on \(\LMod _R\) is left complete (Corollary 8.2.5), these are further equivalent to:
- (5)
-
For every \(Q \in \LMod _R^{\heartsuit }\) and every \(i > 0\), we have \(\Ext ^i_R(P,Q) = 0\).
The projective \(R\)-modules admit a simple description as retracts of free \(R\)-modules.
Definition 8.5.4 (Free module). Let \(R\) be an associative ring spectrum. A left \(R\)-module \(M\) is free if \(M\) is a (possibly infinite) coproduct of (unshifted) copies of \(R\), viewed as a left module over itself. A free left \(R\)-module is finitely generated if it is equivalent to a finite coproduct of copies of \(R\).
Warning 8.5.5. This is a special case of the free-module construction from Subsection 19.1.2. There one allows modules of the form \(R \otimes X\) for an arbitrary spectrum \(X\); here \(X = \bigoplus _{i \in I} \S \) is required to be a coproduct of unshifted copies of the sphere spectrum.
Proposition 8.5.6 ([Lurie (2017), Proposition 7.2.2.7]). Let \(R\) be a connective associative ring spectrum, and let \(P\) be a connective left \(R\)-module. The following conditions are equivalent:
- (1)
-
The left \(R\)-module \(P\) is projective.
- (2)
-
There exists a free \(R\)-module \(M\) such that \(P\) is a retract of \(M\).
Proof. Suppose first that \(P\) is projective. We can choose a map of left \(R\)-modules \(p\colon N \to P\), where \(N\) is free, such that the induced map \(\pi _0(p)\colon \pi _0(N) \to \pi _0(P)\) is surjective. (For instance, we can take \(N\) to be the free module generated by the set \(\pi _0(P)\).) Letting \(N'\) be the fiber of \(p\), it follows from the long exact sequence on homotopy groups that \(N'\) is connective as well. The exact sequence \(N' \to N \to P\) of connective \(R\)-modules induces by Observation 6.4.10 a sequence \[ \Ext ^0_R(P,N') \to \Ext ^0_R(P,N) \to \Ext ^0_R(P,P) \to \Ext ^1_R(P,N'). \] Since \(P\) is projective, the last term vanishes by criterion (3) of Observation 8.5.3. In particular, there exists a map \(s\colon P \to N\) such that \(p \circ s = \id _P\), exhibiting \(P\) as a retract of the free \(R\)-module \(N\).
For the converse, we observe that the collection of projective left \(R\)-modules is stable under retracts. It will therefore suffice to show that every free left \(R\)-module is projective. We check criterion (2) from Observation 8.5.3. Since \(\Ext ^i_R(\bigoplus _I R, Q) \cong \prod _{I} \Ext ^i_R(R,Q)\), we may assume \(P = R\) is free on a single generator. But then \[ \Ext ^i_R(R,Q) = \pi _{0} \Hom _{\LMod _R}(R,Q[i]) \simeq \pi _0 \Hom _{\Sp }(\S ,Q[i]) = \pi _{-i}(Q). \] Since \(Q\) is connective, this is zero for \(i > 0\), as desired. □
Remark 8.5.7. If \(R\) is an ordinary associative ring, it follows from the proposition that the projective \(HR\)-modules are precisely the Eilenberg–MacLane modules \(HP\) associated with projective \(R\)-modules in the classical sense.
8.5.2 Flat modules
We now turn to flat modules. Recall from Definition 6.5.1 the general notion of t-flatness with respect to a tensor product \(C \times D \to E\) of stable \(\infty \)-categories with t-structures. We will specialize this to the relative tensor product of modules over a ring spectrum.
Definition 8.5.8 (Tor-groups for modules). Let \(R\) be an associative ring spectrum. For a right \(R\)-module \(M\) and a left \(R\)-module \(N\), we define the \(n\)-th Tor-group of \(M\) and \(N\) over \(R\) to be \[ \Tor ^R_n(M,N) \quad := \quad \pi _n(M \otimes _R N) \qin \Ab . \] These are the Tor-groups from Definition 6.5.6 specialized to the relative tensor product \(\RMod _R \times \LMod _R \xrightarrow {\otimes _R} \Sp \).
Observation 8.5.9. Let \(R\) be a connective associative ring spectrum. By Observation 8.2.7, the relative tensor product of a connective right \(R\)-module \(M\) and a connective left \(R\)-module \(N\) is connective, and \[ \Tor ^R_0(M,N) = \pi _0(M \otimes _R N) \cong \pi _0(M) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N). \] In particular, \(\Tor ^R_n(M,N) = 0\) for \(n < 0\) whenever \(M\) and \(N\) are connective.
Observation 8.5.10 (Tor long exact sequence). Specializing Observation 6.5.7, every exact sequence of right \(R\)-modules induces a long exact sequence of Tor-groups after tensoring with a fixed left \(R\)-module. The same holds for exact sequences in the left-module variable.
Definition 8.5.11 (Flat module). Let \(R\) be a connective associative ring spectrum. A connective left \(R\)-module \(N\) is said to be flat if the functor \((-) \otimes _R N \colon \RMod _R \to \Sp \) is t-exact.
Remark 8.5.12. By Observation 8.5.9, the functor \((-) \otimes _R N\) is automatically right t-exact when \(N\) is connective. Thus a connective left \(R\)-module \(N\) is flat if and only if this functor is left t-exact: for every coconnective right \(R\)-module \(M\), the spectrum \(M \otimes _R N\) is coconnective.
We now give several complementary tests for flatness. Conditions (2) and (3) describe the homotopy groups of a flat module and of its tensor products, while conditions (4) and (5) reduce flatness to discrete modules. The final two conditions recover the classical criterion for an ordinary ring.
Theorem 8.5.13 (Characterization of flat modules, [Lurie (2017), Proposition 7.2.2.13, Theorem 7.2.2.15]). Let \(R\) be a connective associative ring spectrum and let \(N\) be a connective left \(R\)-module. The following conditions are equivalent:
- (1)
- (2)
-
For every right \(R\)-module \(M\) and \(n \in \Z \), the following map is an isomorphism: \[ \pi _n(M) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \iso \pi _n(M \otimes _R N), \qquad \quad [x] \otimes [y] \mapsto [x \otimes y]. \]
- (3)
-
The discrete \(\pi _0(R)\)-module \(\pi _0(N)\) is flat, and for each integer \(n\) the map \[ \pi _n(R) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \to \pi _n(N), \qquad \qquad [r] \otimes [x] \mapsto [rx] \] is an isomorphism of abelian groups.
- (4)
-
For every discrete right \(R\)-module \(M\), the spectrum \(M \otimes _R N\) is discrete. Equivalently, \(\Tor ^R_i(M,N) = 0\) for all discrete \(M\) and all \(i > 0\).
- (5)
-
Regarding \(\pi _0(R)\) as a discrete ring spectrum, the left \(\pi _0(R)\)-module \(\pi _0(R) \otimes _R N\) is discrete and flat.
If \(R\) and \(N\) are both discrete, these conditions are further equivalent to:
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 \(M\). □
Observation 8.5.14. Let \(R\) be a connective associative ring spectrum. Condition (3) of Theorem 8.5.13 immediately gives the following facts:
- (a)
-
The subcategory of \(\LMod _R\) spanned by the flat \(R\)-modules is closed under coproducts, retracts, and filtered colimits.
- (b)
-
Every free module is flat. Combining this with (a) and Proposition 8.5.6, it follows that every projective module is flat.
- (c)
-
If \(A\) is an ordinary associative ring, then a connective left \(HA\)-module \(N\) is flat if and only if \(N\) is discrete and \(\pi _0 N\) is a flat \(A\)-module in the classical sense, i.e., satisfies condition (6) of Theorem 8.5.13.
Lemma 8.5.15. Let \(R \in \CAlg (\Sp _{\geq 0})\) be a connective commutative ring spectrum, and let \(M\) and \(N\) be flat \(R\)-modules. Then \(M \otimes _R N\) is again flat.
Proof. By Corollary 8.2.5, connectivity and coconnectivity of an \(R\)-module are detected on its underlying spectrum. A connective \(R\)-module \(P\) is therefore flat if and only if the functor \((-) \otimes _R P\colon \Mod _R \to \Mod _R\) is t-exact, as opposed to merely its composite with the forgetful functor to \(\Sp \). Now associativity of the relative tensor product provides a natural isomorphism \[ (-) \otimes _R (M \otimes _R N) \quad \simeq \quad \bigl ((-) \otimes _R M\bigr ) \otimes _R N \] of functors \(\Mod _R \to \Mod _R\). Since \(M\) and \(N\) are flat, the right-hand side is a composite of two t-exact functors, hence t-exact. Finally, \(M \otimes _R N\) is connective by Observation 8.5.9, so it is flat. □
Observation 8.5.16. Every free \(R\)-module is flat (Observation 8.5.14), so in particular the monoidal unit \(R\) is flat. Combined with Lemma 8.5.15, this shows that the full subcategory \(\Mod _R^{\flat } \subseteq \Mod _{R,\geq 0}\) spanned by the flat \(R\)-modules contains the unit and is closed under the tensor product. It therefore inherits a symmetric monoidal structure by the Part II result Lemma 14.5.2.
For connective ring spectra, flat modules admit a characterization analogous to Lazard’s theorem in ordinary algebra. It expresses them as filtered colimits of the simplest flat modules, the finitely generated free ones.
Theorem 8.5.17 (Lazard’s theorem for spectra, [Lurie (2017), Theorem 7.2.2.15]). Let \(R\) be a connective associative ring spectrum and let \(N\) be a connective left \(R\)-module. The following conditions are equivalent:
- (1)
-
The left \(R\)-module \(N\) can be obtained as a filtered colimit of finitely generated free left \(R\)-modules.
- (2)
-
The left \(R\)-module \(N\) can be obtained as a filtered colimit of finitely generated projective left \(R\)-modules.
- (3)
-
The left \(R\)-module \(N\) is flat.
Proof sketch. The implication \((1) \Rightarrow (2)\) is immediate because every free module is projective. The implication \((2) \Rightarrow (3)\) follows from Observation 8.5.14: projective modules are flat, and flat modules are closed under filtered colimits. The reverse implication is the substantive content of [Lurie (2017), Theorem 7.2.2.15], which we use as a black box. Its proof shows that the \(\infty \)-category of maps \(F \to N\), with \(F\) finitely generated and free, is filtered and that the canonical map from the colimit of this diagram to \(N\) is an isomorphism. The filteredness is ultimately reduced to the classical Lazard theorem over \(\pi _0(R)\). □
8.5.3 Tor-amplitude
We conclude with two boundedness properties of perfect modules. The first is immediate from their construction out of the unit.
Lemma 8.5.18. Let \(R\) be a connective associative ring spectrum and let \(M\) be a perfect \(R\)-module in the sense of Definition 8.1.7. Then \(M\) is bounded below: there exists an integer \(n\) such that \(\pi _i(M) = 0\) for all \(i < n\).
Proof. The ring spectrum \(R\) is bounded below, and the bounded below \(R\)-modules form a thick subcategory of \(\LMod _R\): they are closed under shifts, cofibers, and retracts. The claim follows from the definition of a perfect module. □
Tor-amplitude gives a uniform bound after tensoring with discrete modules. By Theorem 8.5.13, flat modules are precisely the connective modules of Tor-amplitude contained in \([0,0]\).
Definition 8.5.19 (Tor-amplitude). Let \(R\) be a connective associative ring spectrum and let \(M\) be a left \(R\)-module. We say that \(M\) has Tor-amplitude contained in \([a,b]\) for integers \(a \leq b\) if for every discrete right \(R\)-module \(N\), the homotopy groups of \(N \otimes _R M\) are concentrated in degrees \([a,b]\), that is, \(\Tor _i^R(N,M) = 0\) whenever \(i < a\) or \(i > b\). If such \(a\) and \(b\) exist, \(M\) is said to have finite Tor-amplitude.
The last three statements below make this bound effective for perfect modules. A perfect module concentrated in a single Tor degree is a shifted finitely generated projective module; in general, one can successively remove the bottom Tor degree.
Proposition 8.5.20 ([Antieau and Gepner (2014), Proposition 2.13]). Let \(R\) be a connective associative ring spectrum and let \(M\) and \(N\) be left \(R\)-module spectra.
- (1)
-
If \(M\) has Tor-amplitude contained in \([a,b]\), then the shift \(M[k]\) has Tor-amplitude contained in \([a + k, b + k]\) for all \(k \in \Z \).
- (2)
-
If \(M\) is perfect, then \(M\) has finite Tor-amplitude.
- (3)
-
If \(R\) happens to be a commutative ring spectrum, \(S \in \CAlg (\Sp )_{R/}\) is connective, and \(M\) has Tor-amplitude contained in \([a, b]\), then the left \(S\)-module \(S \otimes _R M\) has Tor-amplitude contained in \([a, b]\).
- (4)
-
If \(M\) and \(N\) have Tor-amplitude contained in \([a, b]\), then the fiber and cofiber of a map \(M \to N\) have Tor-amplitude contained in \([a - 1, b]\) and \([a, b + 1]\).
- (5)
-
If \(M\) is perfect with Tor-amplitude contained in \([0, b]\), then \(M\) is connective and \(\pi _0 M \cong \pi _0(\pi _0(R) \otimes _R M)\), where \(\pi _0(R)\) is regarded as a discrete ring spectrum.
- (6)
-
If \(M\) is perfect with Tor-amplitude contained in \([a, a]\), then \(M\) is equivalent to \(P[a]\) for a finitely generated projective left \(R\)-module \(P\).
- (7)
-
If \(M\) is perfect with Tor-amplitude contained in \([a, b]\), then there exists an exact sequence \(P[a] \to M \to Q\) with \(P\) finitely generated projective and \(Q\) perfect with Tor-amplitude contained in \([a + 1, b]\).
Proof sketch. Part (1) follows directly from the definition. For (2), observe that \(R\) has Tor-amplitude contained in \([0,0]\) and that the left \(R\)-modules of finite Tor-amplitude form a thick subcategory of \(\LMod _R\): they are closed under shifts and retracts, while closure under cofibers follows from the long exact sequence of Tor-groups (Observation 8.5.10). For (3), if \(T\) is a discrete right \(S\)-module, then \[ T \otimes _S (S \otimes _R M) \simeq T \otimes _R M. \] Part (4) follows from the same long exact sequence of Tor-groups. Parts (5)–(7) require the Tor spectral sequence together with the lifting theory of projective modules, so we take them directly from Antieau and Gepner (2014), Proposition 2.13. □
Exercises
Exercise 8.1 (Perfect modules and base change). Let \(f\colon R\to S\) be a morphism of associative ring spectra.
- (1)
-
Show that extension of scalars \(S\otimes _R-\colon \LMod _R\to \LMod _S\) sends perfect \(R\)-modules to perfect \(S\)-modules.
- (2)
-
Assume that \(R\) and \(S\) are commutative and let \(x\in \pi _0(R)\). Write \(R/x\) for the cofiber of multiplication by \(x\) on \(R\). Show that \(R/x\) is perfect and that \[ S\otimes _R(R/x)\cong S/f(x). \]
Exercise 8.2 (A relative mapping-spectrum calculation). Let \(m,n\geq 1\) and put \(d:=\gcd (m,n)\). Compute \[ \pi _*\hom _{H\Z }\bigl (H(\Z /m),H(\Z /n)\bigr ). \] Show that the only nonzero groups are isomorphic to \(\Z /d\) in degrees \(0\) and \(-1\).
Exercise 8.3 (A relative tensor-product calculation). Let \(m,n\geq 1\) and put \(d:=\gcd (m,n)\). Compute \[ \pi _*\bigl (H(\Z /m)\otimes _{H\Z }H(\Z /n)\bigr ). \] Show that the only nonzero groups are isomorphic to \(\Z /d\) in degrees \(0\) and \(1\).
Exercise 8.4 (Projectivity and flatness of the discrete truncation). Let \(R\) be a connective associative ring spectrum, and regard \(H\pi _0(R)\) as a left \(R\)-module via the canonical morphism \(R \to H\pi _0(R)\). Show that the following conditions are equivalent:
- (1)
-
The ring spectrum \(R\) is discrete.
- (2)
-
The left \(R\)-module \(H\pi _0(R)\) is projective.
- (3)
-
The left \(R\)-module \(H\pi _0(R)\) is flat.
Deduce that \(H\Z \) is neither projective nor flat as a module over the sphere spectrum, even though \(\pi _0(H\Z )\) is free of rank one over \(\pi _0(\S )\).
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