This section explains how ordinary rings and differential graded algebras give rise to ring spectra. We begin with the Eilenberg–MacLane functor, using the symmetric monoidality of \(\pi _0\) to obtain the required multiplicative structure.
Corollary 8.3.1. The Eilenberg–MacLane functor \(H\colon \Ab \longrightarrow \Sp _{\geq 0}\) is canonically lax symmetric monoidal, hence induces functors \[ H\colon \Alg (\Ab )\longrightarrow \Alg (\Sp ) \qquadtext {and} H\colon \CAlg (\Ab )\longrightarrow \CAlg (\Sp ). \]
Proof. Since \(H\) is right adjoint to the symmetric monoidal functor \(\pi _0\colon \Sp _{\geq 0} \to \Ab \), it follows from Proposition 14.3.6 that \(H\) is canonically lax symmetric monoidal. Every lax symmetric monoidal functor preserves associative and commutative algebra objects, as does the symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \). □
Thus every ordinary associative ring \(A\) determines an associative ring spectrum \(HA\), and every ordinary commutative ring \(A\) determines a commutative ring spectrum \(HA\). In either case, the underlying spectrum is the Eilenberg–MacLane spectrum from Definition 6.2.1.
To incorporate differential graded algebras, we pass through the derived \(\infty \)-category. Let \(R\) be a commutative ring. The derived tensor product equips \(\D (R)\) with a symmetric monoidal structure \[ -\otimes _R^{\bL }-\colon \D (R)\times \D (R)\longrightarrow \D (R) \] whose monoidal unit is \(R[0]\). The \(\infty \)-category \(\D (R)\) is stable and cocomplete, and the derived tensor product preserves colimits separately in both variables. In Proposition 20.2.2 we construct this symmetric monoidal structure as a monoidal Dwyer–Kan localization of \(\Ch (R)\). Its compact objects are the perfect complexes, and Corollary 20.2.5 gives an equivalence \[ \D (R)\simeq \Ind (\Perf (R)). \] In particular, \(\D (R)\) is presentable by Proposition 22.3.6.
We can therefore form the \(\infty \)-categories \[ \Alg (\D (R)) \qquad \text {and}\qquad \CAlg (\D (R)) \] of associative and commutative algebra objects in \(\D (R)\). These objects should be thought of as derived associative and commutative \(R\)-algebras. One source of such objects is provided by differential graded algebras.
Definition 8.3.2 (Differential graded algebra). Let \(R\) be a commutative ring. An associative differential graded \(R\)-algebra, or associative DGA over \(R\), is an associative algebra object in the symmetric monoidal category \(\Ch (R)\) of chain complexes of \(R\)-modules. Concretely, it consists of a chain complex \(A_\bullet \), a unit \(R[0]\to A_\bullet \), and an associative multiplication \[ A_p\otimes _R A_q\longrightarrow A_{p+q},\qquad x\otimes y\longmapsto xy, \] which satisfies the Leibniz rule \[ d(xy)=d(x)y+(-1)^p x d(y) \] for homogeneous \(x\in A_p\) and \(y\in A_q\).
A commutative DGA over \(R\) is a commutative algebra object in \(\Ch (R)\). Equivalently, it is an associative DGA whose multiplication is graded commutative: \[ xy=(-1)^{pq}yx \] for homogeneous \(x\in A_p\) and \(y\in A_q\).
Example 8.3.3. Every associative \(R\)-algebra \(A\) defines an associative DGA \(A[0]\) concentrated in degree \(0\). If \(A\) is commutative, then \(A[0]\) is a commutative DGA. As an example with non-zero differential, let \(a\in R\) and consider the Koszul DGA \[ K_R(a):=(\Lambda _R(e),d),\qquad |e|=1,\quad d(e)=a. \] Here \(\Lambda _R(e)\) denotes the exterior algebra on one generator, so this is a commutative DGA in homological degrees \(1\) and \(0\).
In Part II we show that the localization functor \(\gamma \colon \Ch (R)\to \D (R)\) admits a lax symmetric monoidal refinement (Corollary 20.2.4). Since lax symmetric monoidal functors preserve algebra objects, it induces functors \[ \Alg (\Ch (R))\longrightarrow \Alg (\D (R)) \qquad \text {and}\qquad \CAlg (\Ch (R))\longrightarrow \CAlg (\D (R)). \] Thus every associative or commutative DGA determines a corresponding derived algebra. It is a somewhat surprising fact that the first functor is essentially surjective [Lurie (2017), Proposition 7.1.4.6], while the second one is so when \(R\) contains the field \(\Q \) of rational numbers [Lurie (2017), Proposition 7.1.4.11].
The universal property of the symmetric monoidal structure on spectra recalled in Example 8.1.2 now produces a unique colimit-preserving symmetric monoidal functor \[ L_R\colon \Sp \longrightarrow \D (R). \] It sends the sphere spectrum to the monoidal unit \(R[0]\). Since \(\D (R)\) and \(\Sp \) are presentable, Theorem 22.2.5 gives a right adjoint to \(L_R\). The universal enrichment of stable \(\infty \)-categories in spectra identifies this right adjoint as \[ H_R\colon \D (R)\longrightarrow \Sp , \qquad H_R(A)=\hom _{\D (R)}(R[0],A). \] By Proposition 14.3.6, the right adjoint \(H_R\) has a canonical lax symmetric monoidal structure. It consequently induces functors \[ \Alg (\D (R))\longrightarrow \Alg (\Sp ) \qquad \text {and}\qquad \CAlg (\D (R))\longrightarrow \CAlg (\Sp ). \] For \(R=\Z \), the underlying functor of \(H_R\) is the Eilenberg–MacLane functor from Definition 6.2.1, since both are defined by the formula \(A\mapsto \hom _{\D (\Z )}(\Z [0],A)\).
The following comparison is an application of the monogenic Morita theorem from Part II. This theorem identifies a stable category generated by its monoidal unit with modules over the endomorphism ring spectrum of that unit.
Corollary 8.3.4 (Eilenberg–MacLane modules). For every commutative ring \(R\), there is a symmetric monoidal equivalence \[ \D (R)\xrightarrow {\ \simeq \ }\Mod _{HR}(\Sp ). \] Under this equivalence, the functor \(H_R=\hom _{\D (R)}(R[0],-)\) agrees with the forgetful functor \(\Mod _{HR}\to \Sp \), and the lax symmetric monoidal structure on \(H_R\) sends the unit \(R[0]\) to the Eilenberg–MacLane ring spectrum \(HR\) from Corollary 8.3.1.
Proof. Equip \(\D (R)\) with the symmetric monoidal structure of Proposition 20.2.2. Its unit \(R[0]\) is a compact generator by Corollary 20.2.5. Moreover, \[ \pi _n\hom _{\D (R)}(R[0],R[0])\cong H_n(R[0]) \] by Corollary 6.4.5. This is \(R\) for \(n=0\) and zero otherwise. Thus the resulting commutative endomorphism ring spectrum is concentrated in degree \(0\). The adjunction \(\pi _0\colon \Sp _{\geq 0}\rightleftarrows \Ab \noloc H\) is symmetric monoidal by Proposition 8.2.6, Corollary 8.3.1. The induced localization of commutative algebras from Part II (Example 14.5.5), together with the fully faithful symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \), identifies commutative ring spectra concentrated in degree \(0\) with ordinary commutative rings. The multiplication on \(\pi _0\) of the endomorphism ring spectrum is composition of endomorphisms of \(R[0]\), hence the ordinary multiplication of \(R\). It is therefore the Eilenberg–MacLane ring spectrum \(HR\). The result now follows from Theorem 19.5.6. The description of \(H_R\) is built into the comparison functor in that theorem. □
Definition 8.3.5 (Eilenberg–MacLane ring spectrum). Let \(A\) be an associative DGA over a commutative ring \(R\). Its Eilenberg–MacLane ring spectrum is the associative ring spectrum \[ HA:=H_R(\gamma A)\in \Alg (\Sp ). \] If \(A\) is a commutative DGA, the lax symmetric monoidal structure on \(H_R\circ \gamma \) refines \(HA\) to a commutative ring spectrum.
Remark 8.3.6 (Agreement of the Eilenberg–MacLane constructions). For an ordinary associative \(R\)-algebra \(A\), the ring spectrum \(H_R(A[0])\) is canonically equivalent to the direct Eilenberg–MacLane ring spectrum \(HA\) from Corollary 8.3.1. Indeed, both are concentrated in degree \(0\), and their multiplications induce the ordinary multiplication of \(A\) on \(\pi _0\). The classification of associative ring spectra concentrated in degree \(0\) from Corollary 14.5.4 therefore identifies them. If \(A\) is commutative, the same argument gives an equivalence of commutative ring spectra.
Corollary 8.3.7. Let \(R\) be a commutative ring and let \(M\) and \(N\) be \(R\)-modules, regarded as complexes concentrated in degree \(0\). Write \(\Tor _n^R(M,N)\) and \(\Ext _R^n(M,N)\) for their classical Tor- and Ext-groups. Equip \(HM\) and \(HN\) with their natural \(HR\)-module structures. Under the equivalence of Corollary 8.3.4, there are natural isomorphisms \[ \Tor ^R_n(M,N)\cong \pi _n(HM\otimes _{HR}HN) \qquadtext {and}\qquad \Ext _R^n(M,N)\cong \pi _{-n}\hom _{HR}(HM,HN), \]
Proof. The symmetric monoidal equivalence \(\D (R)\simeq \Mod _{HR}\) identifies \(M[0]\otimes _R^{\bL }N[0]\) with \(HM\otimes _{HR}HN\) and the derived hom with \(\hom _{HR}(HM,HN)\). Their homotopy groups compute the classical derived functors by the resolution formulas of Corollary 6.6.28, Proposition 6.4.14. □
Important examples include \(H\Z \), \(H\Q \), and \(H\F _2\). Their multiplicative structures underlie the products in ordinary cohomology and will be used later in the discussion of orientations, Thom isomorphisms, Poincaré duality, and the Chern character.
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