Section 20.2: The symmetric monoidal derived category | Stable homotopy theory and higher algebra

We finish this chapter by applying the preceding machinery to unbounded chain complexes. Recall from Theorem 6.1.30 that the derived \(\infty \)-category of any abelian category is stable. The construction of the monoidal structure requires the following standard structure on unbounded chain complexes, which we use as a black box.

Proposition 20.2.1 (Unbounded projective cofibrations, [Lurie (2017), Propositions 7.1.2.8 and 7.1.2.11]). Let \(R\) be a commutative ring and let \(W\) be the class of quasi-isomorphisms in \(\Ch (R)\). There is a class \(I_{\proj }\) of projective cofibrations with the following properties:

(1): The triple \((\Ch (R),W,I_{\proj })\) is a homotopy cocomplete \(\infty \)-category with weak equivalences and cofibrations.
(2): The unit \(R[0]\) is cofibrant, and a finite tensor product of cofibrant complexes is again cofibrant.
(3): If \(P\) is cofibrant, then \(P\otimes _R-\) is homotopy cocontinuous and preserves quasi-isomorphisms.

The bounded-below projective cofibrations of Proposition 6.6.15 admit a direct construction from ordinary projective resolutions. The unbounded statement is subtler, which is why we refer to Lurie (2017). Its relevance here is that properties (2) and (3), together with Remark 20.1.4, give exactly the left derivability required in Definition 20.1.3.

Proposition 20.2.2 (The symmetric monoidal derived category). Let \(R\) be a commutative ring. The derived \(\infty \)-category \[ \D (R):=\Ch (R)[\{\textup {quasi-isomorphisms}\}^{-1}] \] admits a canonical symmetric monoidal structure whose tensor product is the derived tensor product and whose unit is \(R[0]\). The localization functor \[ \gamma \colon \Ch (R)\longrightarrow \D (R) \] admits a canonical lax symmetric monoidal refinement. Moreover, the derived tensor product preserves small colimits separately in both variables.

Proof. Equip \(\Ch (R)\) with the weak equivalences and projective cofibrations of Proposition 20.2.1. Properties (2) and (3) of that proposition show that the ordinary tensor product is left derivable. The first two assertions therefore follow from Theorem 20.1.6. The description of cocartesian transport in part (3) of Proposition 20.1.14 identifies tensor product in the localization with the total left derived functor of ordinary tensor product, so this is the derived tensor product. Applying the same description to the unique span from \(\emptyset \) to \(\lra {1}\) identifies the monoidal unit with the image of the ordinary unit \(R[0]\).

It remains to prove the assertion about colimits. By part (1) of Corollary 2.2.9, we may represent an object \(X\in \D (R)\) by a cofibrant complex \(P\). For a complex \(Y\), choose a cofibrant replacement \(Q\xrightarrow {\sim }Y\). The preceding identification and part (3) of Proposition 20.2.1 give \[ X\otimes _R^{\bL }\gamma (Y) \simeq \gamma (P\otimes _R Q) \simeq \gamma (P\otimes _R Y). \] Thus \(X\otimes _R^{\bL }-\) is the functor on localizations induced by \(P\otimes _R-\). The latter is homotopy cocontinuous by part (3) of Proposition 20.2.1. The dual form of Theorem 2.2.11 therefore shows that \(X\otimes _R^{\bL }-\) preserves small colimits. By symmetry, the same holds in the other variable. □

Proposition 20.2.3 (Bounded-below comparison). Let \(R\) be a commutative ring. The full subcategory \(\D ^-(R)\subseteq \D (R)\) of bounded below objects is closed under the derived tensor product and contains the monoidal unit. It therefore inherits a symmetric monoidal structure from \(\D (R)\) for which the inclusion \[ \D ^-(R)\lhook \joinrel \longrightarrow \D (R) \] is symmetric monoidal. This structure agrees with the symmetric monoidal structure on \(\D ^-(R)\) constructed in Example 20.1.7.

Proof. The unit \(R[0]\) is bounded below. Let \(X,Y\in \D ^-(R)\), and choose bounded below projective resolutions \(P\to X\) and \(Q\to Y\). Since projective \(R\)-modules are flat, Lemma 6.6.27 shows that tensoring with either \(P\) or \(Q\) preserves quasi-isomorphisms.

Choose cofibrant replacements \(U\xrightarrow {\sim }P\) and \(V\xrightarrow {\sim }Q\) in the unbounded projective cofibration structure of Proposition 20.2.1. The composite \[ U\otimes _R V\longrightarrow U\otimes _R Q\longrightarrow P\otimes _R Q \] is a quasi-isomorphism: this follows for the first map from part (3) of Proposition 20.2.1, and for the second from the preceding observation about \(Q\). Hence the derived tensor product of \(X\) and \(Y\) in \(\D (R)\) is represented by the bounded below complex \(P\otimes _R Q\). Thus \(\D ^-(R)\) is closed under the ambient derived tensor product, and Lemma 14.5.2 gives the asserted restricted symmetric monoidal structure.

It remains to compare this restriction with the bounded-below construction. The inclusion \(\Ch ^-(R)\hookrightarrow \Ch (R)\) is symmetric monoidal. Composing it with the lax symmetric monoidal localization \(\Ch (R)\to \D (R)\) and applying part (4) of Theorem 20.1.6 to the bounded-below localization produces a lax symmetric monoidal refinement of the inclusion \(\D ^-(R)\hookrightarrow \D (R)\). On bounded below projective resolutions \(P\) and \(Q\), its binary structure map is represented by the composite quasi-isomorphism \(U\otimes _RV\to P\otimes _RQ\) displayed above, and is therefore an isomorphism. Its unit map is likewise the identity of \(R[0]\). The refinement is consequently symmetric monoidal, which identifies the bounded-below structure with the restriction of the unbounded one. □

Corollary 20.2.4 (DGAs as derived algebras). The lax symmetric monoidal localization of Proposition 20.2.2 induces functors \[ \Alg (\Ch (R))\longrightarrow \Alg (\D (R)), \qquad \CAlg (\Ch (R))\longrightarrow \CAlg (\D (R)). \]

Proof. Lax symmetric monoidal functors preserve associative and commutative algebra objects. □

Thus every associative or commutative DGA determines an associative or commutative algebra object in \(\D (R)\), respectively.

We finally record the compact presentation of the derived category.

Corollary 20.2.5 (Perfect complexes). The object \(R[0]\) is a compact generator of \(\D (R)\). If \[ \Perf (R):=\D (R)^{\omega } \] denotes the full subcategory of compact objects, then \(\Perf (R)\) is the smallest thick subcategory of \(\D (R)\) containing \(R[0]\). Its objects are precisely the retracts of bounded complexes of finitely generated free \(R\)-modules, and there is an equivalence of symmetric monoidal \(\infty \)-categories \[ \D (R)\simeq \Ind (\Perf (R)). \] Here the symmetric monoidal structure on the right is the one supplied by Theorem 22.5.4.

Proof. Since \(R\) is projective as an \(R\)-module, Corollary 6.4.5 gives \[ \pi _n\hom _{\D (R)}(R[0],X)\cong H_n(X) \] for every \(X\in \D (R)\). The canonical t-structure on \(\D (R)\) is compatible with filtered colimits by [Lurie (2017), Proposition 1.3.5.21], so each homology functor preserves filtered colimits. Since homotopy groups of spectra preserve filtered colimits by Lemma 4.4.28, the displayed formula shows that \(R[0]\) is compact. Its shifts jointly detect zero objects, since a complex is zero in \(\D (R)\) precisely when all its homology groups vanish. It follows from Proposition 22.3.4 that \(R[0]\) is a compact generator.

By Proposition 22.3.4, the compact objects form the smallest thick subcategory containing \(R[0]\); in particular, \(\Perf (R)\) is small. Bounded complexes of finitely generated free modules are obtained from shifts of \(R[0]\) by finite cofiber sequences, using their finite filtrations by degree. Conversely, their retracts form a thick subcategory. This gives the asserted explicit description of \(\Perf (R)\).

The underlying equivalence \(\D (R)\simeq \Ind (\Perf (R))\) now follows from Proposition 22.3.7. The derived tensor product restricts to \(\Perf (R)\): this follows from its exactness separately and the description of \(\Perf (R)\) as the thick subcategory generated by the unit. The uniqueness assertion in Theorem 22.5.4 upgrades the equivalence to a symmetric monoidal equivalence. □

These results provide the higher-algebra input for the comparison with Eilenberg–MacLane modules. Indeed, the Ind-presentation and Proposition 22.3.6 show that \(\D (R)\) is presentable. It is therefore a presentably symmetric monoidal stable \(\infty \)-category with compact generator \(R[0]\). Applying the monogenic Morita theorem, Theorem 19.5.6, and computing the endomorphism ring spectrum of \(R[0]\) gives the symmetric monoidal equivalence \[ \D (R)\simeq \Mod _{HR}(\Sp ). \] The computation and its interpretation in terms of Eilenberg–MacLane ring spectra are carried out in Corollary 8.3.4.

Exercises

Exercise 20.1 (A tensor-stable localization). Let \(k\) be a field and let \(W\) be the quasi-isomorphisms in \(\Ch (k)\).

(1): Show that \(X\otimes _k-\) preserves quasi-isomorphisms for every chain complex \(X\).
(2): Construct the symmetric monoidal structure on \(\D (k)\).
(3): Explain why the same argument fails for a general commutative ring \(R\), and give an explicit complex whose tensor product does not preserve a quasi-isomorphism.

Exercise 20.2 (Deriving the tensor product). Let \(R\) be a commutative ring and choose projective cofibrant replacements \(P\xrightarrow {\sim }C\) and \(Q\xrightarrow {\sim }D\).

(1): Show that \(P\otimes _RQ\) represents \(C\otimes _R^{\bL }D\) in \(\D (R)\).
(2): Prove that it is enough to replace only \(C\) by \(P\) if \(P\otimes _R-\) preserves quasi-isomorphisms, and similarly for \(D\).
(3): Check that the two constructions agree.

Exercise 20.3 (A self-intersection calculation). Let \(R\) be a commutative ring and let \(x\in R\) be a non-zero-divisor. Compute \(R/(x)\otimes _R^{\bL }R/(x)\). Show that its homology is \(R/(x)\) in degrees \(0\) and \(1\) and vanishes otherwise. Interpret the degree-\(1\) class as the derived contribution to the self-intersection.

Exercise 20.4 (Derived tensor products of cyclic groups). Let \(m,n\geq 1\) and put \(d:=\gcd (m,n)\). Compute \[ \Z /m\otimes _{\Z }^{\bL }\Z /n. \] Show that its only nonzero homology groups occur in degrees \(0\) and \(1\), and that both are isomorphic to \(\Z /d\).

Exercise 20.5 (The lax monoidal localization map). Let \(\gamma \colon \Ch (R)\to \D (R)\) be the localization functor. Write down the lax symmetric monoidal comparison \[ \gamma (C)\otimes _R^{\bL }\gamma (D)\longrightarrow \gamma (C\otimes _RD). \] Show that it is an isomorphism if either \(C\otimes _R-\) or \(-\otimes _R D\) preserves quasi-isomorphisms, and give an example in which it is not an isomorphism.

Exercise 20.6 (Duality for a perfect complex). Let \(R\) be a commutative ring and let \(P\) be a bounded complex of finitely generated projective \(R\)-modules. Define the dual complex by \[ P^{\vee }:=\uHom _R(P,R). \] Show that \(P^{\vee }\) is perfect and that for every \(M\in \D (R)\) there is a natural isomorphism \[ P^{\vee }\otimes _R^{\bL }M\simeq \bR \uHom _R(P,M). \]

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