The cotangent complex and Thom spectra

We first prove, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then prove the following theorem: if $R$ is an $E_\infty$-ring spectrum and $f:G\to \mathrm{Pic}(R)$ is a map of $E_\infty$-groups, then the cotangent complex over $R$ of the Thom $E_\infty$-$R$-algebra of $f$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.


INTRODUCTION
1.1. 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 infintesimal thickening of a commutative R-algebra S is given by a square-zero extension: that is, a surjective map of commutative R-algebras φ : S → S such that the product of any two elements in the kernel of φ is zero.
One way to construct square-zero extensions is using derivations. An R-linear derivation S → M is a map of R-modules which satisfies the Leibniz condition. The S-module of R-linear derivations Der R (S, M) is co-represented by the S-module of Kähler differentials Ω S/R . It is also represented by the commutative R-algebra over S given by the trivial square-zero extension S ⊕ M with multiplication given by (s, m)(s , m ) = (ss , sm + ms ). Therefore, we have bijections The pullback S gets a commutative R-algebra structure, and the map S → S is surjective with kernel isomorphic to M as an R-module. However, this does not produce all square-zero extensions: for that, one needs to derive the module of Kähler differentials. The way this was originally achieved independently by André [And67] and Quillen [Qui68] was by simplicial methods. Quillen placed a model structure on the category of simplicial commutative A-algebras. This produces a B-module LΩ S/R (called the algebraic cotangent complex by Lurie [Lur18b,25.3]), and the first André-Quillen cohomology group Ext 1 S (LΩ S/R , M) is isomorphic to the S-module of equivalence classes of square-zero extensions of S by M.
We shall not take this approach, but rather work with the more general E ∞ -ring spectra. The module of Kähler differentials in this context is replaced by the cotangent complex: if A → B is a map of E ∞ -ring spectra, the cotangent complex is a B-module L B/A . Using L S/R 1 instead of LΩ S/R , André-Quillen (co)homology is replaced by topological André-Quillen (co)homology, also known as TAQ. The precise relation between LΩ B/A and L B/A can be found in [Lur18b,25.3.3.7/25.3.5.4].
One can define square-zero extensions of E ∞ -ring spectra. The cotangent complex helps classify them. As is usual in homotopy theory, one does not merely want to describe equivalence classes of square-zero extensions. One would like to prove that the whole ∞-category of squarezero extensions of an E ∞ -A-algebra B is equivalent to the ∞-category of A-linear derivations B → ΣM where M is a B-module. 2 This is proven in [Lur17,7.4.1.26], provided one restricts the ∞-categories a bit. Note that the traditional definition of derivations as linear maps that satisfy the Leibniz rule does not work in this setup, whereas the interpretations of (1.1) do.
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 [Lur17,7.4.3.4]. 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 [Lur17, 7.5.0.6].
1.2. 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 [Kri93]. It was defined as a sequential colimit built out of tensoring B ∧ A B with spheres, in a certain way.
Basterra [Bas99] 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 [BM05] 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 [Bec03], 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 Ω ∞ Σ ∞ 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. [Sch11,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 Section 3. 1 When treating a discrete commutative ring as an E ∞ -ring spectrum, the Eilenberg-Mac Lane functor shall be understood.
2 Recall that Ext 1 The approach to the cotangent complex taken by Lurie in [Lur17,7.3] 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 [Lur17, Chapter 6]. Our Section 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. [Kuh07,5.4].
In summary, we prove in the ∞-categorical context of [Lur17] 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(Ω ∞ Σ ∞ (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 (3.22) • As the module of indecomposables of the augmentation ideal of B ∧ A B (3.30).
Here −⊗− is a certain operation to be introduced in Notation 3.20 that takes a pointed space and an E ∞ -A-algebra B and returns a B-module.
1.3. 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 [ABG + 14]. 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 : 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 [ABG18], [ACB19]. Examples of Thom E ∞ -ring spectra include complex cobordism MU and periodic complex cobordism MUP. Theorem 4.3. Let R be an E ∞ -ring spectrum. Let f : G → Pic(R) be a map of E ∞ -groups. 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 [BM05]. For example, we recover the equivalence of MU-modules L MU MU ∧ bu of that paper. The result of Basterra and Mandell, however, only applies to Thom spectra of maps to BGL 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 MUP MUP ∧ ku as MUP-modules. In fact, L MUP 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 Section 5 we extend Theorem 4.3 to cotangent complexes of étale extensions of Thom algebras: if M f → B is an étale map of E ∞ -R-algebras, then This allows us, for example, to recover the equivalence L KU KU ∧ HQ from [Sto19].
1.4. Notation and conventions. We will freely use the language of ∞-categories as developed in [Lur09], [Lur17].
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 C B//B the ∞-category (C /B ) id B / .
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 Σ ∞ + , and if G ∈ CAlg(S), then S[G] denotes the E ∞ -ring spectrum Σ ∞ + (G).

SHORT REVIEW OF 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 [Lur17, 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).

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 by this functor, and it is denoted L B/A . If A is an initial object of C, then L B/A is also denoted L B and it is called the absolute cotangent complex of B.
Remark 3.2. Let us say a word about the Σ ∞ + functor above. Since id : B → B is the terminal object of C /B , then 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.
3.1. 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 ∞ -B-algebras: 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 Remark 3.4. The functor I is right adjoint to the free E ∞ -B-algebra functor M → n≥0 (M ∧ B n ) Σ n where (−) Σ n denotes the (homotopy) orbits for the Σ n -action [Lur17, 3.1.3.14], which is augmented over B via the projection to the 0-th summand [Lur17, 7.3.4.5]. Note that I is reduced, as it takes B to the zero module.
Remark 3.6. The augmentation ideal functor factors through NUCA B as follows: is an equivalence of ∞-categories; in particular, Remark 3.8. A model-categorical precedent can be found as Theorem 3 of [BM05]. 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 ΣX n → X n+1 . Spectra in M have a model structure whose fibrant objects are the Ω-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 ΣX n → X n+1 get sent to ΣF(X n ) F(ΣX n ) → F(X n+1 ). After passing to the underlying ∞-categories, I gives a functor equivalent to ∂I.
On the other hand, note that ∂I typically does not commute with Σ ∞ . If it did, then which is the excisive approximation of I by 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 If f : A → B ∈ CAlg, then L B/A ∈ Sp(CAlg B//B ) by definition. Given Theorem 3.7, in this Therefore, by 2.(6), L B/A is equivalently the value of an excisive approximation to I : 3.2. 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 [Lur17,7.3.5] Lurie analyzes this particular case. He denotes by L [Lur17,7.3.5.6] 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 Section 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 2.(3), Here e n : I(Σ n C) → ΩI(Σ n+1 C) is the natural map obtained as in [Lur17, 1.4.2.12]. 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(Σ 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 [RSV19, Section 2] for more details.
Remark 3.14. The suspension ΣA 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, Σ n A S n A for all n ≥ 0.
Notation 3.15. Let B denote the tensor of CAlg B//B over S * . Let ⊗ A denote the tensor of CAlg A over S.
Remark 3.16. If f : A → B in CAlg and (X, x 0 ) ∈ S * , then X ⊗ A B ∈ CAlg B//B , with unit and augmentation given by In the following lemma, we shall prove a couple of results about this construction.
(1) There is a functor which preserves colimits separately in each variable and extends − ⊗ A − : S × CAlg A → CAlg A , in the sense that it makes the following diagram commute: Here the two vertical maps are forgetful functors.
(1) Since the tensor is a colimit and colimits in overcategories are created in the original ∞-category [Lur09, 1.2.13.8], the functor − ⊗ A − : S × CAlg A → CAlg A begets a functor It is defined on objects as follows: where the equivalences A ∅ ⊗ and acts on objects as follows: Now recall from Remark 3.2 that (CAlg A//B ) * CAlg B//B . This equivalence just disregards the redundant upper part of the squares, and gives us the functor we were looking for.
(2) To prove − ⊗ A B and − B (B ∧ A B) are equivalent, it suffices to see that they send S 0 to equivalent objects. Indeed, they are colimit-preserving functors from S * to a pointed presentable ∞-category, but S * is freely generated under colimits by S 0 in pointed presentable ∞-categories [RSV19, 2.29]. Now, indeed both functors send S 0 to B ∧ A B, and the proof is finished. Proof. Consider the following commutative diagram in Mod B :  From the two lemmas above, we deduce: Letting C = B ∧ A B and using Remark 3.14, we can now recast (3.12) in a different form: There is an equivalence of B-modules where Ω denotes the loop functor in Mod B .
Remark 3.23. The previous proposition was known to the experts (it is mentioned e.g in [Sch11, 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 above.
Remark 3.24. Let B be an E ∞ -A-algebra. Forgetting structure, we may consider B as an E k -Aalgebra, 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 , so by Proposition 3.22 we obtain an equivalence of B-modules recovering [Lur17, 7.3.5.6].
3.3. 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 [Bas99]. 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 Section 4. (1) The functor Q is left adjoint to a functor Z : Mod B → NUCA B such that UZ id Mod B , and for each M ∈ Mod B the multiplication map ZM ∧ B ZM → ZM is zero.
(2) Q • F id Mod B , where F : Mod B → NUCA B is the free functor.
(3) There exists a unique functor Z : Mod B → NUCA B such that UZ id Mod B , up to equivalence.
Here U : NUCA B → Mod B denotes the forgetful functor.

Proof.
(1) We will use a criterion for adjointness from [ In the following theorem, we prove how to get the cotangent complex via indecomposables. A model categorical precedent of the result can be found in [BM05,Theorem 4].
Theorem 3.28. Let B ∈ CAlg. The following diagram commutes: Proof. Recall from 2.(6) that P 1 I, the excisive approximation to I, is equivalent to Ω ∞ • ∂I • Σ ∞ . Thus, we have to prove that Q • I 0 is equivalent to P 1 I. Recall that I U • I 0 . By [Lur17, 6.1.1.30], we have P 1 I P 1 U • I 0 . We will now prove that P 1 U Q, finishing the proof.
By 2.(6), Note that Q is excisive, since it preserves pushouts and Mod B is stable, so Q P 1 Q. Therefore, to prove that P 1 U Q it suffices to prove that ∂U ∂Q, by 2.(6) once more.  Remark 3.29. 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 by 2.(6), and Σ ∞ , Ω ∞ are equivalences since Mod B is stable. Using the analogy of Goodwillie calculus with classical calculus, we can gain some intuition for this result. Under this analogy, functors correpond to smooth functions of the real line, so the functor U • F which maps M ∈ Mod B to n≥1 (M ∧ B n ) Σ n corresponds to the power series f (x) = ∑ ∞ n=1 x n . The linear approximation at 0 of this function is the identity map x → x. Continuing with the analogy, linear approximations of functions correspond to 1-excisive approximations of functors, which provides some intuition for the equivalence P 1 (U • F) id Mod B .
From Theorem 3.28 we immediately deduce: Corollary 3.30. Let f : A → B be a map of E ∞ -ring spectra. There is an equivalence of B-modules This is analogous to what Basterra [Bas99] adopted as a definition for L B/A in a modelcategorical setting. That was the first published definition. The approach from (3.10) had been used in the preprint [Kri93], albeit formulated in a different language. Only in [BM05] 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 [Lur17, 2.1.3.1], or, equivalently, a commutative monoid object in S in the sense of [Lur17, 2.4.2.1] (i.e. a special Γ-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  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 Σ ∞ Ω ∞ X → X, in such a way that the main assertion will have been proven, by 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 , 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 * (Ω ∞ Σ n X, −) corresponds to the left fibration (S * ) Ω ∞ Σ n X/ → S * . Pulling back this left fibration along the functor Ω ∞ Σ n : Sp → S * , gives us the left fibration that classifies the functor Map S * (Ω ∞ Σ n X, Ω ∞ Σ n −). Taking limits, it follows that the functor lim n Map S * (Ω ∞ Σ n X, Ω ∞ Σ n −) corresponds to the left fibration lim n (Sp × S * (S * ) Ω ∞ Σ n X/ ) Sp × lim N S * lim n (S * ) Ω ∞ Σ n X/ . By the Yoneda lemma, a map of left fibrations Sp X/ → Sp × lim N S * lim n (S * ) Ω ∞ Σ n X/ is uniquely determined by a choice of object in the fiber of Sp × lim N S * lim n (S * ) Ω ∞ Σ n X/ → Sp over X, which is given by the space {X} × lim n Map S * (Ω ∞ Σ n X, Ω ∞ Σ n X). Thus the object (X, (id Ω ∞ Σ 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(· · · ]. This implies that the top horizontal map is an equivalence as well [Lur09, 3.3.1.3]. Thus, in order to prove that I is an equivalence, by 2-out-of-3 it suffices to show that π 2 • I : Sp X/ → lim n (S * ) Ω ∞ Σ n X/ is an equivalence.
First, we will construct an equivalence Sp X/ − → lim n (S * ) Ω ∞ Σ 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 Ω ∞ Σ 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.
Combining all three gives us a natural equivalence Map Sp (X, −) − → Map Sp (colim n Ω n Σ ∞ Ω ∞ Σ n X, −) which by the Yoneda lemma is induced by a map colim n Ω n Σ ∞ Ω ∞ Σ 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 Ω n Σ ∞ Ω ∞ Σ n X → X can be characterized as the unique map that comes from the cocone given by Ω n Σ ∞ Ω ∞ Σ n X → Ω n Σ n X − → X where the first arrow is induced by the counit of the (Σ ∞ , Ω ∞ ) adjunction.
Theorem 4.3. Let R be an E ∞ -ring spectrum. Let f : G → Pic(R) be a map of E ∞ -groups. There is an equivalence of M f -modules whereas by the definition of J in Lemma 3.19 we get the equivalence (4.4).
By Proposition 3.22 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 Ω 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 more formally as where U n : Grp E ∞ (S) → Grp E n (S), E 0 -groups are pointed spaces, and i n : S ≥n * → S * is the inclusion functor of (n − 1)-connected pointed spaces. Now note that B ∞ is constructed as the limit of the B n as follows [Lur17,5.2.6.26] id y y so in particular we have a commutative diagram where the top horizontal functor is the projection to the corresponding term of the limit. After composing it with i n this functor becomes Ω ∞−n = Ω ∞ Σ 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. Proposition 4.9. Let R be an E ∞ -ring spectrum, let G be an E ∞ -ring space and let f : G → Pic(R) be an E ∞ -map (with respect to the E ∞ -group structure of G). Then the R-module L M f /R underlies a Thom E ∞ -(R ∧ B ∞ G)-algebra.
Proof. First, note that if R → T is a map of E ∞ -ring spectra, then extension of scalars induces 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 Tmodules. Now recall that, as an R-module, M f  Example 4.11. Continuing the example of MUP from Example 4.8, if we take f to be the complex J-homomorphism BU × Z → Pic(S), then L MUP MUP ∧ 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 [AHL09].
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 superflous: 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 HG is the Eilenberg-Mac Lane spectrum of G, and that since G is a discrete group then Σ ∞ + G is a wedge of sphere spectra. In particular, π 0 (L M f ) = G G. 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. − → G f − → Pic(R))).

ÉTALE EXTENSIONS
We will now extend the results of the previous section to compute cotangent complexes of étale extensions of Thom E ∞ -algebras.
Following [Lur17, 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 L B/R B ∧ A L A/R .
Proof. The first statement is [Lur17,7.5.4.5]. 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 [Lur17,7.3 Example 5.4. Let KU denote the periodic complex topological K-theory E ∞ -ring spectrum.