The cotangent complex and Thom spectra

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-ring spectra in various ways. In this work we first establish, in the context of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-ring spectrum and Pic(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Pic}(R)$$\end{document} denote its Picard E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-group. Let Mf denote the Thom E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-R-algebra of a map of E∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\infty $$\end{document}-groups f:G→Pic(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:G\rightarrow \mathrm {Pic}(R)$$\end{document}; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of R→Mf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\rightarrow Mf$$\end{document} is equivalent to the smash product of Mf and the connective spectrum associated to G.


Deformation theory
The cotangent complex arises as a central object of deformation theory. One wishes to understand extensions of functions between geometric spaces to infinitesimal thickenings of those spaces. Working algebraically, the simplest example of an infinitesimal thickening of a commutative R-algebra S is given by a square-zero extension: that is, a surjective map of  Apart from the connections to deformation theory, the cotangent complex L B/A helps detect useful properties of the map f : A → B. For example, if A and B are connective, then f is an equivalence if and only if π 0 ( f ) : π 0 (A) → π 0 (B) is an isomorphism and L B/A vanishes [16, 7.4.3.4]. The theory of the cotangent complex also helps in proving theorems that do not mention it at all: for example, if A is an E ∞ -ring spectrum and a map of commutative rings π 0 (A) → B 0 is étale, then it lifts in an essentially unique way to an étale map of E ∞ -ring spectra A → B [16, 7.5.0.6].

Different approaches to the cotangent complex
Let us trace the history of the E ∞ cotangent complex of a map of E ∞ -ring spectra A → B, since it has been defined in different ways in the literature.
The first definition can be found in a preprint by Kriz [13]. It was defined as a sequential colimit built out of tensoring B ∧ A B with spheres, in a certain way.
Basterra [6] took another approach: she defined the cotangent complex to be the indecomposables of the augmentation ideal I (B ∧ A B) of B ∧ A B, which is the fiber of the multiplication map B ∧ A B → B. She did not prove the equivalence with Kriz's approach.
Later, herself and Mandell [8] established the connection between Basterra's definition of the cotangent complex and stabilization. Just as Beck had observed in the sixties that the category of modules over a commutative ring R was equivalently given by the abelian group objects in augmented commutative R-algebras [7], they proved the E ∞ -analog. Abelian group objects have to be replaced by spectra objects. They proved that to get L B/A , one can start from B ∧ A B considered as an augmented commutative B-algebra, then stabilize it, i.e. apply ∞ ∞ to it, then take its augmentation ideal.
It was known to the experts that one could extract from the results of Basterra and Mandell an expression of the cotangent complex as a sequential colimit, similar to Kriz's expression, see e.g. [21,Page 164]. However, we think a full description of how these approaches are connected has not appeared in the literature. We take the opportunity to expand on them in Sect. 3.
The approach to the cotangent complex taken by Lurie in [16, 7.3] is closest to Basterra and Mandell's approach, albeit in the realm of ∞-categories rather than in that of model categories. We feel a unified study of the different approaches in Lurie's setting was lacking: we provide one here. We adopt the language of the Goodwillie calculus of functors as developed by Lurie in [16,Chapter 6]. Our Sect. 2 will swiftly introduce the necessary results. The fact that the the cotangent complex can be understood via Goodwillie calculus was known, see e.g. [14, 5.4].
In summary, we prove in the ∞-categorical context of [16] that the cotangent complex L B/A can be presented in the following ways: • As the augmentation ideal of the stabilization of B ∧ A B, i.e. I ( ∞ ∞ (B ∧ A B)) (3.9/3.10), • As the excisive approximation of I evaluated in B ∧ A B, i.e. (P 1 I )(B ∧ A B) (3.11), • As the sequential colimit of B-modules (3.26) • As the module of indecomposables of the augmentation ideal of B ∧ A B (3.34).
Here −⊗− is a certain operation to be introduced in Notation 3.24 that takes a pointed space and an E ∞ -A-algebra B and returns a B-module.

Thom spectra
The main result of this paper is the determination of the cotangent complex of Thom E ∞ -ring spectra. Let us quickly recall what these are, following the ∞-categorical approach of [1]. Let G be a space, R be an E ∞ -ring spectrum, Pic(R) be the Picard space of R (the subspace of Mod R spanned by the invertible R-modules), and f : G → Pic(R) be a map. The colimit When G is an E ∞ -group and f is a map of E ∞ -groups, then M f gets the structure of an E ∞ -R-algebra [2,3]. Examples of Thom E ∞ -ring spectra include complex cobordism MU and periodic complex cobordism MU P.
There is an equivalence of M f -modules Here B ∞ G denotes the connective spectrum associated to G.
A model-categorical version had first appeared in [8]. For example, we recover the equivalence of MU -modules of that paper. The result of Basterra and Mandell, however, only applies to Thom spectra of maps to BG L 1 (S), whereas ours applies to maps to Pic(R) where R is any E ∞ -ring spectrum. This allows for generalized, possibly non-connective Thom E ∞ -ring spectra. For example, we get that L MU P MU P ∧ ku as MU P-modules, where ku denotes the E ∞ -ring spectrum of connective complex topological K -theory. In fact, L MU P is actually a Thom E ∞ -ku-algebra. More generally, we observe in Proposition 4.9 that when G is an E ∞ -ring space, then the R-module L M f /R underlies a Thom E ∞ -(R ∧ B ∞ G)-algebra. In other words, with this additional hypothesis the cotangent complex of a Thom E ∞ -algebra becomes a Thom E ∞ -algebra. In Sect. 5 we extend Theorem 4.3 to cotangent complexes of two types of extensions of Thom algebras: if M f → B is a map E ∞ -R-algebras which is either étale or solid (i.e. the multiplication B ∧ M f B → B is an equivalence), then This allows us, for example, to recover the equivalence L K U K U ∧ H Q from [24], where K U denotes the E ∞ -ring spectrum of periodic complex topological K -theory.

Notation and conventions
We will freely use the language of ∞-categories as developed in [15,16].
Let C be an ∞-category. Given a fixed map f : A → B, we denote by C A/ /B the ∞category (C /B ) f / of objects C ∈ C together with maps A → C → B which compose to f . Similarly, we denote by If C is an ∞-category with terminal object T , we let C * denote the undercategory C T / . The ∞-category of spaces will be denoted by S , and that of spectra by Sp. Its full subcategory of connective spectra is denoted by Sp cn . We denote by CAlg the ∞-category CAlg(Sp) of E ∞ -ring spectra, and by CAlg R that of E ∞ -algebras over an E ∞ -ring spectrum R, i.e. CAlg(Mod R ). The suspension spectrum functor S → Sp is denoted ∞ + , and if G ∈ CAlg(S ), then S[G] denotes the E ∞ -ring spectrum ∞ + (G).

Short review of stabilization and Goodwillie calculus
Let us summarize some notions from the Goodwillie calculus of functors which we shall be using. We shall work with the ∞-categorical version of it as in [16,Chapter 6]. For simplicity, let us assume that C and D are ∞-categories which are pointed and presentable.
(1) A functor F : C → D is reduced if it takes a final object to a final object. It is excisive if it takes pushout squares to pullback squares. The full subcategory of Fun(C , D) spanned by the excisive functors is denoted Exc(C , D), and the one spanned by the reduced, excisive functors is denoted Exc * (C , D). (2) The ∞-category of spectra in C is defined by Sp(C ) = Exc * (S fin * , C ) where S fin * is the ∞-category of pointed finite spaces [16, 1.4.2.8]. Evaluation at the sphere S 0 defines a functor ∞ : Sp(C ) → C which is an equivalence when C is stable [16, 1.4.2.21]; since C is pointed and presentable, ∞ admits a left adjoint ∞ : C → Sp(C ) [16, 1.4.4.4]. If C is presentable but not pointed, the left adjoint to ∞ is denoted ∞ + and it factors into two left adjoint functors C [11, 4.10].
(3) The inclusion Exc(C , D) → Fun(C , D) has a left adjoint P 1 , called the excisive approximation functor [16, 6.1.1.10]. The unit natural transformation F ⇒ P 1 F is said to exhibit P 1 F as the excisive approximation to F. If F : C → D is reduced, then P 1 F is reduced, and by [16, 6.1.1.28], In particular, if C is stable then id C colim n ( n C • n C ). (8) If F : C → D is reduced, left exact and preserves filtered colimits, then as follows from (6) and (4).

The cotangent complex
In this section, we introduce the cotangent complex of a map of E ∞ -ring spectra and we give different expressions for it: via the augmentation ideal, via a stabilization process, i.e. as a sequential colimit, and via indecomposables. Before going to E ∞ -ring spectra, let us say a word on the general definition. The relative cotangent complex according to Lurie is a suspension spectrum, in the following sense: Definition 3.1 Let C be a presentable ∞-category and f : A → B in C . Consider the suspension spectrum functor  Note as well that C A/ /B = (C /B ) f / (C A/ ) / f . On the other hand, by [16, 7.3.3.9], we have (C A/ /B ) * C B/ /B . Therefore, ∞ + factors as In order to address the issue of functoriality of the cotangent complex, Lurie uses the tangent bundle of C A/ /B . We shall not be needing this, so for the sake of simplicity we will not introduce it.
Let us now concentrate on the case C = CAlg.

The cotangent complex via the augmentation ideal
When C = CAlg, we may identify Sp(C B/ /B ) with a more familiar ∞-category, namely Mod B , as we shall now see. Note that CAlg B/ /B is the ∞-category of augmented E ∞ -Balgebras: its objects are E ∞ -B-algebras C with a map C → B of E ∞ -B-algebras.

Definition 3.3
Let B ∈ CAlg. The augmentation ideal functor takes C to the fiber in Mod B of the augmentation, i.e. to the pullback is an equivalence of ∞-categories; in particular, Remark 3.8 A model-categorical precedent can be found as Theorem 3 of [8]. There, the functor fitting in the place of ∂ I is defined as follows. First of all, in their framework a spectrum in a model category M is a sequence of objects {X n } n≥0 of M with maps X n → X n+1 . Spectra in M have a model structure whose fibrant objects are the -spectra. Thus, any topological left Quillen functor F between model categories enriched over based spaces induces a left Quillen functor F between the corresponding model categories of spectra: the arrows X n → X n+1 get sent to F(X n ) F( X n ) → F(X n+1 ).
In particular, the augmentation ideal functor from the model category of augmented commutative B-algebras to the model category of B-modules, let us also call it I , induces a functor I between the respective model categories of spectra. After passing to their underlying ∞-categories, I gives a functor equivalent to ∂ I .

Remark 3.9 Recall from Sect. 2.(4) that
On the other hand, note that ∂ I typically does not commute with ∞ . If it did, then but on the other hand this is equivalent to which is the excisive approximation of I by Sect. 2. (8). Therefore, I would be excisive. Since Mod B is stable, this would mean that I preserves pushouts; since I is also reduced, then I would be right exact. But this is typically false. For example, I does not commute with coproducts: if we take B = S, the coproduct of Therefore, by Sect. 2. (6), L B/A is equivalently the value of an excisive approximation to In symbols,

The cotangent complex as a colimit
Let A be an E ∞ -ring spectrum. The general definition of a cotangent complex also applies to an E k -A-algebra B. In [16, 7.3.5] Lurie analyzes this particular case. He denotes by L Forgetting structure, every E ∞ -A-algebra B is an E k -A-algebra for every k ≥ 0. Lurie observes in [16, 7.3.5.6] that since the E ∞ -operad is the colimit of the E k -operads, these E k -cotangent complexes recover the cotangent complex as follows: These E k -cotangent complexes admit a different expression which is sometimes computable, as we shall see in this section. That is what we shall use in Sect. 4 to compute the cotangent complex of Thom E ∞ -algebras.
Let B ∈ CAlg and C ∈ CAlg B/ /B . Since I is left exact (it is a right adjoint) and commutes with sequential colimits (Remark 3.6), then by Sect. 2. (3), (3.12) Here e n : I ( n C) → I ( n+1 C) is the natural map obtained as in [16, 1.4.2.12]. Explicitly, it is obtained as follows. Write n+1 C as the pushout of B ← n C → B (remember that B is a zero object of CAlg B/ /B ). Apply I , then e n is defined as the universal pullback map Let f : A → B be a morphism in CAlg. Applying (3.12) to C = B ∧ A B we get a quite explicit colimit formula for L B/A . But we can be more explicit: we are going to recast I ( n (B ∧ A B)) in other terms.
Any presentable ∞-category C is tensored over spaces: there is a functor − ⊗ − : S × C → C which preserves colimits separately in each variable. If X ∈ S and c ∈ C , then where {c} denotes the constant functor with value c. If C is moreover pointed, then it is tensored over pointed spaces: there is a functor − − : S * × C → C which preserves colimits separately in each variable. If (X , x 0 ) ∈ S * and c ∈ C , then See [20, Sect. 2] for more details.
Remark 3.14 The suspension A of an object A in a pointed presentable ∞-category C can be expressed as S 1 A. Indeed, write S 1 = colim( * ← S 0 → * ) and apply the colimit-preserving functor − A. By induction, n A S n A for all n ≥ 0.

with unit and augmentation given by
We shall now prove a couple of results about this construction.

Remark 3.17
The definition of an adjunction in [15, 5.2], which we are implicitly adopting, uses the theory of correspondences. We shall use the result of Cisinski [10, 6.1.23; Footnote, Page 250] which says that Lurie's definition is equivalent to the expected characterization via natural equivalences of mapping spaces Map C (c, Gd) Map D (Fc, d).
In the following lemma, we shall need the following notation: if F : C → D is a functor of ∞-categories and c ∈ C , we denote by F : C c/ → D F(c)/ the induced functor on undercategories. Note that if F has a right adjoint G, then by Remark 3.17 we conclude that F also has a right adjoint. Indeed, if we are given maps c → c and c → Gd in C , then the natural equivalence restricts to a natural equivalence Explicitly, the right adjoint of F takes an object g : Fc → d to the composition c → G Fc Gg −→ Gd. Therefore, if the ∞-categories are presentable and F preserves colimits, then F also preserves colimits.
We will now consider bifunctors that preserve colimits separately in each variable and analyze in which way this property passes on to undercategories.
For a given h : c → c in C , the functor There is an analogous result in the other variable, starting from an arrow d → d ∈ D, but we shall not be needing it.
Proof We can factor the functor F(h, −) as the composition The first functor preserves colimits by the discussion above applied to F(c , −) : D → E , which preserves colimits by hypothesis. Therefore F(h, −) preserves colimits if and only if F(h, id d ) * preserves colimits.
If F(h, id d ) * preserves colimits, then it preserves initial objects, which forces F(h, id d ) to be an equivalence. The converse is obvious.

Remark 3.19
The functor F of Lemma 3.18 always preserves colimits indexed by weakly contractible simplicial sets separately in each variable, by [15, 4.4.2.9]. We shall not be using this fact, though. (3.21) in the sense that the following diagram commutes, where the two vertical maps are forgetful functors: The functor (3.21) takes (X , C) to X ⊗ A C with unit and augmentation given as in Remark 3.16.
Proof (1) which extends the original one and preserves colimits separately in each variable. Now, note as in Remark 3.2 that We adopt the notation of [13] or [ We can now recast (3.12) in a different form. Define arrows e n : S n ⊗ A B → (S n+1 ⊗ A B) as follows. Start by presenting S n+1 ∈ S * as the pushout of * ← S n → * . Apply − ⊗ A B : S * → CAlg B/ /B to it, and then apply I . Now e n is the universal pullback map S n ⊗ A B → (S n+1 ⊗ A B).
where denotes the loop functor in Mod B .
Proof We use the characterization L B/A (P 1 I )(B ∧ A B) from (3.11). Consider the equivalence (3.12) with C = B ∧ A B. By Remark 3.14 and Corollary 3.25, we have The maps e n from (3.12) and the maps e n are defined in an analogous fashion, so the natural equivalences above commute with them.

Remark 3.27
The previous proposition was known to the experts (it is mentioned e.g in [21, Page 164]), but we do not think a complete derivation had been spelled out in the literature before.
Finally, let us make the connection between the ⊗ construction and the E k -cotangent complex mentioned at the beginning of the subsection.

Remark 3.28
Let B be an E ∞ -A-algebra. Forgetting structure, we may consider B as an E k -A-algebra, for all k ≥ 0, and thus we may form its E k -cotangent complex L where * : S k−1 → * denotes the unique map. Since * ⊗ id : S k−1 ⊗ A B → B is the augmentation of S k−1 ⊗ A B and loops commute with pullbacks, this identifies L [9, 1.3]), so by Proposition 3.26 we obtain an equivalence of B-modules recovering [16, 7.3.5.6].

The cotangent complex via indecomposables
The first published definition of the cotangent complex was established in the context of model categories using the indecomposables functor [6]. The goal of this subsection is to prove that this definition of the cotangent complex is equivalent to the definition adopted in (3.10). We are not aware of a discussion of the approach using indecomposables in the ∞-categorical setting.
The content of this subsection will not be used in the sequel: the reader interested in Thom spectra should feel free to jump ahead to Sect. 4.

Definition 3.29
Let B ∈ CAlg. We denote by the indecomposables functor that takes N to the cofiber in Mod B of the multiplication, i.e. to the pushout The functor Q is left adjoint to the functor which takes a module M and endows it with a zero multiplication map. More precisely: (2) Q • F is the left adjoint to U • Z id Mod B , so it is equivalent to the identity.
(3) The existence of such a Z has just been proven. Now suppose we have a functor Z : We have natural equivalences of spaces Let N ∈ NUCA B . Note that the free-forgetful adjunction (F, U ) is monadic by [16, 4.7.3.5], since NUCA B is an ∞-category of algebras over an ∞-operad in Mod B [16, 5.4 To prove that ∂U is an equivalence, we proceed similarly as in [ To see this, note that Sym n is the composition By [16, 6.1.3.12], this functor is n-reduced, so it is 2-reduced, which by definition means that its excisive approximation is nullhomotopic. 5 By Sect. 2.(6), its derivative is nullhomotopic as well, since Mod B is stable.

Remark 3.33
The key aspect of the previous proof is the fact that ∂(U • F) id Sp(Mod B ) , which is proven in the last paragraph. Notice this is equivalent to From Theorem 3.32 we immediately deduce:

Corollary 3.34 Let f : A → B be a map of E ∞ -ring spectra. There is an equivalence of B-modules
This is analogous to what Basterra [6] adopted as a definition for L B/A in a model-categorical setting. That was the first published definition. The approach from (3.10) had been used in the preprint [13], albeit formulated in a different language. Only in [8] were the two approaches first proven to be equivalent.

The cotangent complex of Thom spectra
In this section we will prove the main result, Theorem 4.3, giving an expression for the cotangent complex of Thom E ∞ -algebras.
An E ∞ -monoid is a commutative algebra object in S in the sense of [16, 2.1.3.1], or, equivalently, a commutative monoid object in S in the sense of [16, 2.4

.2.1] (i.e. a special -space). They form an ∞-category Mon E ∞ (S ).
If an E ∞ -monoid M is grouplike, i.e. if the monoid π 0 (M) is a group, we say M is an E ∞ -group. These form an ∞-category denoted by Grp E ∞ (S ).
Let R be an E ∞ -ring spectrum. An R-module M is invertible if there exists an R-module N such that M ∧ R N R. We let Pic(R) be the Picard space of R: this is the core (i.e. the maximal subspace) of the full subcategory of Mod R on the invertible R-modules.
If Z is a space and f : Z → Pic(R) is a map of spaces, the Thom R-module of f is defined as This defines a functor M : S /Pic(R) → Mod R . As noted in [2, 7.7]

The main result
We will need the following proposition. In particular, for a spectrum X , there is an equivalence natural in X .
Proof We will first establish the natural equivalence (4.2). We will then observe that this equivalence is obtained from the counit ∞ ∞ X → X , in such a way that the main assertion will have been proven, by Sect. 2.(3). Let X be a spectrum. First, we prove there is a natural equivalence of functors where the arrows in the sequential limit are loop functors. We do this carefully, taking care of naturality: indeed, it is easier to see that the two functors are objectwise equivalent using that Sp lim(· · · − → S * − → S * ) [ , it suffices to establish an equivalence of the corresponding left fibrations. The left hand side corresponds to the left fibration Sp X / → Sp. For the right hand side, first observe that for n ∈ N, the functor Map S * ( ∞ n X , −) corresponds to the left fibration (S * ) ∞ n X / → S * . Pulling back this left fibration along the functor ∞ n : Sp → S * , gives us the left fibration that classifies the functor Map S * ( ∞ n X , ∞ n −). Taking limits, it follows that the functor lim n Map S * ( ∞ n X , ∞ n −) corresponds to the left fibration lim n (Sp × S * (S * ) ∞ n X / ) Sp × lim N S * lim n (S * ) ∞ n X / . By the Yoneda lemma, a map of left fibrations Sp X / → Sp × lim N S * lim n (S * ) ∞ n X / is uniquely determined by a choice of object in the fiber of Sp× lim N S * lim n (S * ) ∞ n X / → Sp over X , which is given by the space {X }×lim n Map S * ( ∞ n X , ∞ n X ). Thus the object (X , (id ∞ n X ) n∈N ) in the fiber induces a map of left fibrations over Sp We want to prove this map is an equivalence. Note we can recover the right hand side directly as the following pullback: The bottom equivalence is Sp lim(· · · − → S * − → S * ) = lim N S * , from [16, 1.4.2.24]. This implies that the top horizontal map is an equivalence as well [15, 3.3.1.3]. Thus, in order to prove that I is an equivalence, by 2-out-of-3 it suffices to show that is an equivalence. First, we will construct an equivalence Sp X / − → lim n (S * ) ∞ n X / . Then we will observe it is indeed π 2 • I . We have Using that limits commute with pullbacks and exponentials (−) 1 , gives us an equivalence. Notice this equivalence takes the object id X in Sp X / to (id ∞ n X ) n∈N and thus, by the Yoneda lemma, is equivalent to π 2 • I , as they both take id X to the same object. This gives us the desired result. Next, we have a natural equivalence induced by the adjunction ( ∞ , ∞ ). Finally, we have a natural equivalence Combining all three gives us a natural equivalence which by the Yoneda lemma is induced by a map colim n n ∞ ∞ n X → X and is explicitly given by the image of the identity map id X under this natural equivalence. By inspection, the map colim n n ∞ ∞ n X → X can be characterized as the unique map that comes from the cocone given by n ∞ ∞ n X → n n X − → X where the first arrow is induced by the counit of the ( ∞ , ∞ ) adjunction.
Let us fix some notation necessary for the statement of the following theorem and its proof. Let B ∞ : Grp E ∞ (S ) → Sp cn denote the standard equivalence between the ∞-categories of E ∞ -groups and of connective spectra [16, 5.2.6.26]. Let B : Grp E ∞ (S ) → Grp E ∞ (S ) denote the bar construction functor, which can be defined in this context simply as the functor that takes G to the pushout * ← G → * . We will iterate this functor, getting B n : Grp E ∞ (S ) → Grp E ∞ (S ) for n ≥ 1.
If we take an E 1 -group instead of an E ∞ -group as input, then we can extend the bar construction to a functor Bar : Grp E 1 (S ) → S * [16, 5.2.2]. It agrees with BG whenever this makes sense: namely, if G ∈ Grp E ∞ (S ), then the pointed space underlying BG is equivalent to Bar of the E 1 -group underlying G; this follows from the remarks in [16, 5.2.2.3/4].
Proof By [20, 4.11] as M f -modules. Indeed, since S 0 * + (x 0 ) + (B n G) + B n G is a cofiber sequence in S * where x 0 denotes the basepoint of B n G, then applying M f ∧ ∞ (−) we get a cofiber sequence in whereas by the definition of J in Lemma 3.23 we get the equivalence (4.4). By Proposition 3.26 we get equivalences of M f -modules where in the second line we have used the stability of Sp and Mod M f together with the fact that M f ∧ − : Sp → Mod M f commutes with colimits; note that now denotes the loop functor in Sp.
We will now prove that B ∞ G is equivalent to the colimit in (4.5), which we can rewrite as where U n : Grp E ∞ (S ) → Grp E n (S ) is the forgetful functor, E 0 -groups are pointed spaces, and i n : S ≥n * → S * is the inclusion functor. Now note that B ∞ is constructed as the limit of the Bar (n) as follows [16, 5.2.6.26] (the functors β n in that reference are the inverses of Bar (n) [16, 5.2.6.10(3) where the top horizontal functor is the projection to the corresponding term of the limit. After composing it with i n this functor becomes ∞−n = ∞ n : Sp cn → S * , the n-th space functor. Therefore, (4.6) is equivalent to This is equivalent to B ∞ G by Proposition 4.1.

The cotangent complex as a Thom spectrum
If G is not only an E ∞ -group but also an E ∞ -ring space in the sense of [11, 7.
Proof First, note that if R → T is a map of E ∞ -ring spectra, then extension of scalars restricts to an E ∞ -map − ∧ R T : Pic(R) → Pic(T ) making the following diagram commute: Indeed, the functor − ∧ R T : Mod R → Mod T takes invertible R-modules to invertible T -modules. Now recall that, as an R-module, M f Since we proved in Theorem 4.
this proves the result.

Remark 4.10
This echoes with the result that the factorization homology of a Thom E nalgebra is a Thom spectrum [12, 4.2], or with the related result that the tensor of a Thom E ∞ -algebra with a space is again a Thom E ∞ -algebra [20, 4.10].

Example 4.11
Continuing the example of MU P from Example 4.8, if we take f to be the complex J -homomorphism BU ×Z → Pic(S), then L MU P MU P ∧ku underlies a Thom E ∞ -ku-algebra, even if ku is not the Thom spectrum of any E 3 -map from an E 3 -group [4].
In Proposition 4.9, we proved that L M f /R is the Thom module of an E ∞ -map provided G is an E ∞ -ring space. The hypothesis was not superfluous: let us now give an example of a cotangent complex of a Thom E ∞ -ring spectrum which cannot be the Thom module of an E ∞ -map, simply because it would then be an E ∞ -ring spectrum and this cannot happen in this example:

Example 4.12
Let G be an abelian group of infinite order such that every element has finite order. For example, one could take Q/Z or an infinite direct sum of cyclic groups of arbitrarily large order.
Let f : G → Pic(S) be the the constant map at S. Let us prove that L M f is not the underlying spectrum of an E ∞ -ring spectrum. Such a structure would induce a ring structure on π 0 (L M f ), so it suffices to prove that the latter has no ring structure extending its abelian group structure.
Before we prove this, let us use Theorem 4.3 to compute L M f . We have equivalences of spectra where we used the fact that B ∞ G H G is the Eilenberg-Mac Lane spectrum of G, and that since G is a discrete group then ∞ + G is a wedge of sphere spectra. In particular, Now, note that G = G G is again an abelian group of infinite order such that every element has finite order. This implies that it is not the underlying abelian group of a ring. If it were, the multiplicative unit would have finite order, and this number would bound the order of all the other elements, contradicting that G has infinite order.

Remark 4.13
As we just observed, in general L M f /R is not the Thom module of an E ∞ -map. It is, however, the colimit of iterated loops of Thom modules of E ∞ -maps by (4.5), since by [20, 4.8] and so

Étale extensions and solid ring spectra
We will now extend the results of the previous section to compute cotangent complexes of two different types of extensions of Thom E ∞ -algebras.
Following [16, 7.5.0.1/2/4], a map of (ordinary) commutative rings A → B is étale if B is finitely presented as an A-algebra, B is flat as an A-module, and there exists an idempotent element e ∈ B ⊗ A B such that the multiplication map B ⊗ A B → B induces an isomorphism (B ⊗ A B)[e −1 ] ∼ = B. If A → B is a map of E ∞ -ring spectra, it is étale if π 0 (A) → π 0 (B) is étale and B is flat as an A-module, i.e. the natural map π * (A) ⊗ π 0 (A) π 0 (B) → π * (B) is an isomorphism.

Proposition 5.1 If A → B is an étale map of E ∞ -ring spectra, then L B/A vanishes. Therefore, if R is an E ∞ -ring spectrum and A → B is a map of E ∞ -R-algebras which is étale, there is an equivalence of B-modules
The first statement is [16, 7.5.4.5]. Note that Lurie adds a connectivity hypothesis, but it is not used in the proof. The second statement now follows from the transitivity cofiber sequence [16, 7.3  There are interesting instances where we want to invert a homotopy element x that is not in degree 0, as we shall see below. Unfortunately, in this case the map R → R[x −1 ] may not be étale: indeed, it may not be flat. For example, if R is connective then any flat R-module is necessarily connective, as follows from the definition [16, 7.2.2.11].
To remedy this, we recall the notion of solidity: An E ∞ -A-algebra B is solid if the multiplication map μ : B ∧ A B → B is an equivalence. Note in this case B ∧ A B B as objects of CAlg B/ /B .

Proposition 5.4 If A is an E ∞ -ring spectrum and B is a solid E ∞ -A-algebra, then L B/A vanishes. Therefore, if R is an E ∞ -ring spectrum, A an E ∞ -R-algebra and B a solid E ∞ -A-algebra, then there is an equivalence of B-modules
Proof For the first statement, consider the equivalences where the first equivalence is (3.11) and the second follows from solidity. Now (P 1 I )(B) is trivial, since B ∈ CAlg B/ /B is a zero object and P 1 I is reduced (see Sect. 2. (3)). The second statement now follows from the transitivity cofiber sequence [16, 7.3.3.6] B ∧ A L A/R → L B/R → L B/A .
The following corollary is proven analogously to Corollary 5.2.

Corollary 5.5 Let R be an E
This generalizes Example 5.3, where x was only allowed to be in degree zero.

Example 5.7
Let K U denote the periodic complex topological K -theory E ∞ -ring spectrum.
Since S[K (Z, 2)] M(K (Z, 2) {S} − → Pic(S)), we can apply Corollary 5.5 and Example 5.6 to deduce that L K U K U ∧ 2 H Z. Recall that the inclusion Z → Q induces an equivalence K U ∧ H Z K U ∧ H Q [25, 16.25]. Combining this result with Bott periodicity, we obtain: the rationalization of K U , a result first gotten in [24, 8.4] in a model-categorical context.