In this section, we introduce the cotangent complex of a map of \(E_\infty \)-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 \(E_\infty \)-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 \(\mathscr {C}\) be a presentable \(\infty \)-category and \(f:A\rightarrow B\) in \(\mathscr {C}\). Consider the suspension spectrum functor
The relative cotangent complex of f is the image of \(A\xrightarrow {f}B\xrightarrow {\mathrm {id}}B\) by this functor, and it is denoted \(L_{B/A}\). If A is an initial object of \(\mathscr {C}\), 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 \(\Sigma ^\infty _+\) functor above. Since \(\mathrm {id}:B\rightarrow B\) is the terminal object of \(\mathscr {C}_{/B}\), then
$$\begin{aligned}\mathscr {C}_{{{B}/ / {B}}}=(\mathscr {C}_{/B})_{\mathrm {id}_B/} \simeq (\mathscr {C}_{/B})_*.\end{aligned}$$
Note as well that \(\mathscr {C}_{{{A}/ / {B}}}=(\mathscr {C}_{/B})_{f/}\simeq (\mathscr {C}_{A/})_{/f}\). On the other hand, by [16, 7.3.3.9], we have \((\mathscr {C}_{{{A}/ / {B}}})_* \simeq \mathscr {C}_{{{B}/ / {B}}}\). Therefore, \(\Sigma ^\infty _+\) factors as
In order to address the issue of functoriality of the cotangent complex, Lurie uses the tangent bundle of \(\mathscr {C}_{{{A}/ / {B}}}\). We shall not be needing this, so for the sake of simplicity we will not introduce it.
Let us now concentrate on the case \(\mathscr {C}=\mathrm {CAlg}\).
The cotangent complex via the augmentation ideal
When \(\mathscr {C}=\mathrm {CAlg}\), we may identify \(\mathrm {Sp}(\mathscr {C}_{{{B}/ / {B}}})\) with a more familiar \(\infty \)-category, namely \(\mathrm {Mod}_B\), as we shall now see. Note that \(\mathrm {CAlg}_{{{B}/ / {B}}}\) is the \(\infty \)-category of augmented \(E_\infty \)-B-algebras: its objects are \(E_\infty \)-B-algebras C with a map \(C\rightarrow B\) of \(E_\infty \)-B-algebras.
Definition 3.3
Let \(B\in \mathrm {CAlg}\). The augmentation ideal functor
$$\begin{aligned} I:\mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {Mod}_B \end{aligned}$$
takes C to the fiber in \(\mathrm {Mod}_B\) of the augmentation, i.e. to the pullback
in \(\mathrm {Mod}_B\), where 0 denotes a zero object of \(\mathrm {Mod}_B\).
Remark 3.4
The functor I is right adjoint to the functor \(\mathrm {Mod}_B \rightarrow \mathrm {CAlg}_{{{B}/ / {B}}}\) which takes a module M to the free \(E_\infty \)-B-algebra \(\bigvee _{n\ge 0} (M^{\wedge _B n})_{\Sigma _n}\in \mathrm {CAlg}_B\) endowed with the augmentation over B given by projection to the 0-th summand. Here \((-)_{\Sigma _n}\) denotes the (homotopy) orbits for the \(\Sigma _n\)-action [16, 3.1.3.14, 7.3.4.5]. Note that I is reduced, as it takes B to the zero module.
Notation 3.5
We let \(\mathrm {NUCA}_B\) denote the \(\infty \)-category of non-unital \(E_\infty \)-B-algebras, which we call nucasFootnote 3 [16, 5.4.4.1].
Remark 3.6
The augmentation ideal functor factors through \(\mathrm {NUCA}_B\) as follows:
where U is the forgetful functor. Indeed, one can take the pullback defining I in Definition 3.3 in the \(\infty \)-category \(\mathrm {NUCA}_B\) instead of in \(\mathrm {Mod}_B\), which defines \(I_0\). The functor \(I_0\) is a right adjoint to the functor \(N\mapsto B\vee N\), and it is in fact an equivalence [16, 5.4.4.10], [6, 2.2]. Note as well that I commutes with sifted colimits, since U does [16, 3.2.3.1]. In particular, I commutes with filtered or sequential colimits.
Theorem 3.7
[16, 7.3.4.7/14] The functor
is an equivalence of \(\infty \)-categories; in particular, \(\mathrm {Sp}(\mathrm {CAlg}_{{{B}/ / {B}}}) \xrightarrow {\partial I} \mathrm {Sp}(\mathrm {Mod}_B) \xrightarrow {\Omega ^\infty } \mathrm {Mod}_B\) is an equivalence as well.
Remark 3.8
A model-categorical precedent can be found as Theorem 3 of [8]. There, the functor fitting in the place of \(\partial I\) is defined as follows. First of all, in their framework a spectrum in a model category \(\mathcal {M}\) is a sequence of objects \(\{X_n\}_{n\ge 0}\) of \(\mathcal {M}\) with maps \(\Sigma X_n\rightarrow X_{n+1}\). Spectra in \(\mathcal {M}\) have a model structure whose fibrant objects are the \(\Omega \)-spectra. Thus, any topological left Quillen functor F between model categories enriched over based spaces induces a left Quillen functor \({\underline{F}}\) between the corresponding model categories of spectra: the arrows \(\Sigma X_n \rightarrow X_{n+1}\) get sent to \(\Sigma F(X_n)\simeq F(\Sigma X_n) \rightarrow F(X_{n+1})\).
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 \({\underline{I}}\) between the respective model categories of spectra. After passing to their underlying \(\infty \)-categories, \({\underline{I}}\) gives a functor equivalent to \(\partial I\).
Remark 3.9
Recall from Sect. 2.(4) that \(\Omega ^\infty _{\mathrm {Mod}_B} \circ \partial I \simeq I\circ \Omega ^\infty _{\mathrm {CAlg}_{{{B}/ / {B}}}}\), i.e. \(\partial I\) commutes with \(\Omega ^\infty \). On the other hand, note that \(\partial I\) typically does not commute with \(\Sigma ^\infty \). If it did, then
$$\begin{aligned} \Omega ^\infty _{\mathrm {Mod}_B} \circ \partial I \circ \Sigma ^\infty _{\mathrm {CAlg}_{{{B}/ / {B}}}} \simeq \Omega ^\infty _{\mathrm {Mod}_B} \circ \Sigma ^\infty _{\mathrm {Mod}_B} \circ I \simeq I, \end{aligned}$$
but on the other hand this is equivalent to \(I \circ \Omega ^\infty _{\mathrm {CAlg}_{{{B}/ / {B}}}} \circ \Sigma ^\infty _{\mathrm {CAlg}_{{{B}/ / {B}}}}\) which is the excisive approximation of I by Sect. 2.(8). Therefore, I would be excisive. Since \(\mathrm {Mod}_B\) 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 \(B=\mathbb {S}\), the coproduct of \(\mathbb {S}[S^1]\) with itself in \(\mathrm {CAlg}_{{{\mathbb {S}}/ / {\mathbb {S}}}}\) is \(\mathbb {S}[S^1\times S^1]\). Its augmentation ideal is \(\Sigma ^\infty (S^1\times S^1)\simeq \Sigma ^\infty S^1\vee \Sigma ^\infty S^1 \vee \Sigma ^\infty S^2\), which is not equivalent to the coproduct of \(I(\mathbb {S}[S^1])\simeq \Sigma ^\infty S^1\) with itself.
If \(f:A\rightarrow B\in \mathrm {CAlg}\), then \(L_{B/A}\in \mathrm {Sp}(\mathrm {CAlg}_{{{B}/ / {B}}})\) by definition. Given Theorem 3.7, in this situation we redefine \(L_{B/A}\) to mean the image of \(A\xrightarrow {f}B\xrightarrow {\mathrm {id}}B\) under the composition
Therefore, by Sect. 2.(6), \(L_{B/A}\) is equivalently the value of an excisive approximation to \(I:\mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {Mod}_B\) evaluated in \(B\wedge _A B\). In symbols,
$$\begin{aligned} L_{B/A}\simeq (P_1I)(B\wedge _A B). \end{aligned}$$
(3.11)
The cotangent complex as a colimit
Let A be an \(E_\infty \)-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 \(L^{(k)}_{B/A}\) the resulting \(E_k\)-cotangent complex.
Forgetting structure, every \(E_\infty \)-A-algebra B is an \(E_k\)-A-algebra for every \(k\ge 0\). Lurie observes in [16, 7.3.5.6] that since the \(E_\infty \)-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 \(E_\infty \)-algebras.
Let \(B\in \mathrm {CAlg}\) and \(C\in \mathrm {CAlg}_{{{B}/ / {B}}}\). Since I is left exact (it is a right adjoint) and commutes with sequential colimits (Remark 3.6), then by Sect. 2.(3),
Here \(e_n:I(\Sigma ^nC) \rightarrow \Omega I(\Sigma ^{n+1} C)\) is the natural map obtained as in [16, 1.4.2.12]. Explicitly, it is obtained as follows. Write \(\Sigma ^{n+1}C\) as the pushout of \(B\leftarrow \Sigma ^n C \rightarrow B\) (remember that B is a zero object of \(\mathrm {CAlg}_{{{B}/ / {B}}}\)). Apply I, then \(e_n\) is defined as the universal pullback map \( I(\Sigma ^n C)\rightarrow \Omega I(\Sigma ^{n+1}C)\).
Let \(f:A\rightarrow B\) be a morphism in \(\mathrm {CAlg}\). Applying (3.12) to \(C=B\wedge _A B\) we get a quite explicit colimit formula for \(L_{B/A}\). But we can be more explicit: we are going to recast \(I(\Sigma ^n (B\wedge _A B))\) in other terms.
Any presentable \(\infty \)-category \(\mathscr {C}\) is tensored over spaces: there is a functor \(-\otimes -:\mathscr {S}\times \mathscr {C}\rightarrow \mathscr {C}\) which preserves colimits separately in each variable. If \(X\in \mathscr {S}\) and \(c\in \mathscr {C}\), then
$$\begin{aligned} X\otimes c\simeq \mathrm {colim}(X\xrightarrow {\{c\}} \mathscr {C}) \end{aligned}$$
where \(\{c\}\) denotes the constant functor with value c. If \(\mathscr {C}\) is moreover pointed, then it is tensored over pointed spaces: there is a functor \(-\odot -:\mathscr {S}_*\times \mathscr {C}\rightarrow \mathscr {C}\) which preserves colimits separately in each variable. If \((X,x_0)\in \mathscr {S}_*\) and \(c\in \mathscr {C}\), then
$$\begin{aligned} X\odot c \simeq \mathrm {cofib}_\mathscr {C}(c\simeq *\otimes c {\mathop {\longrightarrow }\limits ^{x_0 \otimes \mathrm {id}}} X\otimes c). \end{aligned}$$
(3.13)
See [20, Sect. 2] for more details.
Remark 3.14
The suspension \(\Sigma A\) of an object A in a pointed presentable \(\infty \)-category \(\mathscr {C}\) can be expressed as \(S^1\odot A\). Indeed, write \(S^1=\mathrm {colim}(*\leftarrow S^0 \rightarrow *)\) and apply the colimit-preserving functor \(-\odot A\). By induction, \(\Sigma ^nA \simeq S^n\odot A\) for all \(n\ge 0\).
Notation 3.15
Let \(\odot _B\) denote the tensor of \(\mathrm {CAlg}_{{{B}/ / {B}}}\) over \(\mathscr {S}_*\). Let \(\otimes _A\) denote the tensor of \(\mathrm {CAlg}_A\) over \(\mathscr {S}\).
Remark 3.16
If \(f:A\rightarrow B\) in \(\mathrm {CAlg}\) and \((X,x_0)\in \mathscr {S}_*\), then \(X\otimes _A B\in \mathrm {CAlg}_{{{B}/ / {B}}}\), with unit and augmentation given by
$$\begin{aligned} B\simeq *\otimes _A B \xrightarrow {x_0\otimes \mathrm {id}} X\otimes _A B \xrightarrow {*\otimes \mathrm {id}} *\otimes _A B \simeq B. \end{aligned}$$
More generally, if \((B\xrightarrow {g} C\xrightarrow {e} B) \in \mathrm {CAlg}_{{{B}/ / {B}}}\), then \(X\otimes _A C\in \mathrm {CAlg}_{{{B}/ / {B}}}\), with unit and augmentation given by
$$\begin{aligned} B\simeq *\otimes _A B \xrightarrow {x_0\otimes g} X\otimes _A C \xrightarrow {*\otimes e} *\otimes _A B \simeq B. \end{aligned}$$
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 \(\mathrm {Map}_\mathscr {C}(c,Gd) \simeq \mathrm {Map}_\mathscr {D}(Fc,d)\).
In the following lemma, we shall need the following notation: if \(F:\mathscr {C}\rightarrow \mathscr {D}\) is a functor of \(\infty \)-categories and \(c\in \mathscr {C}\), we denote by \({\overline{F}}:\mathscr {C}_{c/}\rightarrow \mathscr {D}_{F(c)/}\) the induced functor on undercategories. Note that if F has a right adjoint G, then by Remark 3.17 we conclude that \({\overline{F}}\) also has a right adjoint. Indeed, if we are given maps \(c\rightarrow c'\) and \(c\rightarrow Gd\) in \(\mathscr {C}\), then the natural equivalence
$$\begin{aligned} \mathrm {Map}_\mathscr {C}(c',Gd) \simeq \mathrm {Map}_\mathscr {D}(Fc',d) \end{aligned}$$
restricts to a natural equivalence
$$\begin{aligned} \mathrm {Map}_{\mathscr {C}_{c/}}(c',Gd) \simeq \mathrm {Map}_{\mathscr {D}_{Fc/}}(Fc',d). \end{aligned}$$
Explicitly, the right adjoint of \({\overline{F}}\) takes an object \(g:Fc \rightarrow d\) to the composition \(c \rightarrow GFc \xrightarrow {Gg} Gd\). Therefore, if the \(\infty \)-categories are presentable and F preserves colimits, then \({\overline{F}}\) 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.
Lemma 3.18
Let \(F: \mathscr {C}\times \mathscr {D}\rightarrow \mathscr {E}\) be a functor of presentable \(\infty \)-categories that preserves colimits separately in each variable. Let \((c,d)\in \mathscr {C}\times \mathscr {D}\). Consider the induced functor
$$\begin{aligned} {\overline{F}}:\mathscr {C}_{c/}\times \mathscr {D}_{d/}\rightarrow \mathscr {E}_{F(c,d)/}. \end{aligned}$$
For a given \(h:c \rightarrow c'\) in \(\mathscr {C}\), the functor
$$\begin{aligned} {\overline{F}}(h, - ) : \mathscr {D}_{d/} \rightarrow \mathscr {E}_{F(c,d)/} \end{aligned}$$
preserves colimits if and only if \(F(h,\mathrm {id}_d): F(c,d) \rightarrow F(c',d)\) is an equivalence.
There is an analogous result in the other variable, starting from an arrow \(d\rightarrow d'\in \mathscr {D}\), but we shall not be needing it.
Proof
We can factor the functor \({\overline{F}}(h, - )\) as the composition
$$\begin{aligned} \mathscr {D}_{d/} \xrightarrow { \overline{F(c',-)} } \mathscr {E}_{F(c',d)/} \xrightarrow {F(h,\mathrm {id}_d)_* } \mathscr {E}_{F(c,d)/}. \end{aligned}$$
The first functor preserves colimits by the discussion above applied to \(F(c',-): \mathscr {D}\rightarrow \mathscr {E}\), which preserves colimits by hypothesis. Therefore \({\overline{F}}(h,-)\) preserves colimits if and only if \(F(h,\mathrm {id}_d)_*\) preserves colimits.
If \(F(h,\mathrm {id}_d)_*\) preserves colimits, then it preserves initial objects, which forces \(F(h,\mathrm {id}_d)\) to be an equivalence. The converse is obvious. \(\square \)
Remark 3.19
The functor \({\overline{F}}\) 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.
Proposition 3.20
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(1)
The functor \(- \otimes _A -: \mathscr {S}\times \mathrm {CAlg}_A \rightarrow \mathrm {CAlg}_A\) extends to a functor
$$\begin{aligned} - \otimes _A -: \mathscr {S}_* \times \mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {CAlg}_{{{B}/ / {B}}}\end{aligned}$$
(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 \(X\otimes _A C\) with unit and augmentation given as in Remark 3.16.
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(2)
Letting the second variable of (3.21) be of fixed value \((B \rightarrow C \rightarrow B) \in \mathrm {CAlg}_{{{B}/ / {B}}}\), the restricted functor
$$\begin{aligned} - \otimes _A C: \mathscr {S}_* \rightarrow \mathrm {CAlg}_{{{B}/ / {B}}}\end{aligned}$$
preserves colimits if and only if the unit \(B \rightarrow C\) is an equivalence.
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(3)
Letting the second variable of (3.21) be of fixed value \(B \xrightarrow {\mathrm {id}}B \xrightarrow {\mathrm {id}} B\), the restricted functor
$$\begin{aligned} -\otimes _A B: \mathscr {S}_*\rightarrow \mathrm {CAlg}_{{{B}/ / {B}}}\end{aligned}$$
is equivalent to \(-\odot _B (B\wedge _A B)\), where \(B\wedge _A B\) denotes the object \(B \xrightarrow {\iota _0} B\wedge _AB \xrightarrow {\mu } B\) of \(\mathrm {CAlg}_{{{B}/ / {B}}}\).Footnote 4
Proof
-
(1)
Since the tensor is a colimit and colimits in overcategories are created in the original \(\infty \)-category [15, 1.2.13.8], the functor \(-\otimes _A-:\mathscr {S}\times \mathrm {CAlg}_A\rightarrow \mathrm {CAlg}_A\) begets a functor
$$\begin{aligned} -\otimes _A-:\mathscr {S}\times (\mathrm {CAlg}_A)_{/B} \rightarrow (\mathrm {CAlg}_A)_{/B} \end{aligned}$$
(3.22)
which extends the original one and preserves colimits separately in each variable. Now, note as in Remark 3.2 that
$$\begin{aligned} (\mathrm {CAlg}_A)_{/B} \simeq (\mathrm {CAlg}_{A/})_{/f} \simeq \mathrm {CAlg}_{{{A}/ / {B}}}. \end{aligned}$$
We now consider \((*,A\rightarrow B\rightarrow B)\in \mathscr {S}_*\times \mathrm {CAlg}_{{{A}/ / {B}}}\) and consider the functor induced by (3.22) in undercategories. This gives the result, since \(A\rightarrow B\rightarrow B\) is a terminal object of \(\mathrm {CAlg}_{{{A}/ / {B}}}\), and by Remark 3.2\((\mathrm {CAlg}_{{{A}/ / {B}}})_*\simeq \mathrm {CAlg}_{{{B}/ / {B}}}\). The functor we just obtained acts on objects as expected, by construction.
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(2)
This follows from Lemma 3.18, since the condition there amounts in this case to the map \(* \otimes _A B \rightarrow * \otimes _A C\) being an equivalence.
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(3)
To prove \(-\otimes _A B\) and \(-\odot _B (B\wedge _A B)\) are equivalent, it suffices to see that they send \(S^0\) to equivalent objects. Indeed, they are colimit-preserving functors from \(\mathscr {S}_*\) to a pointed presentable \(\infty \)-category, but \(\mathscr {S}_*\) is freely generated under colimits by \(S^0\) in pointed presentable \(\infty \)-categories [20, 2.29]. Now, indeed both functors send \(S^0\) to \(B\wedge _A B\), and the proof is finished.
\(\square \)
Lemma 3.23
Let \(B \xrightarrow { \ u \ } C \xrightarrow { \ c \ } B\) be an object in \(\mathrm {CAlg}_{{{B}/ / {B}}}\). Then I(C) is naturally equivalent to the cofiber of u in \(\mathrm {Mod}_B\), i.e.
$$\begin{aligned} I(C)=\mathrm {fib}_{\mathrm {Mod}_B}(c:C\rightarrow B) \simeq \mathrm {cofib}_{\mathrm {Mod}_B}(u:B\rightarrow C) {=}{:}J(C). \end{aligned}$$
Proof
Consider the following commutative diagram in \(\mathrm {Mod}_B\):
The bottom left square and the big rectangle formed by the two squares on the left are pullbacks, so the square on the top left is a pullback [15, 4.4.2.1]. It is therefore a pushout, by stability of \(\mathrm {Mod}_B\). The square on the top right is also a pushout, hence the big rectangle formed by the two squares on top is a pushout [15, Dual of 4.4.2.1], proving the result. \(\square \)
We adopt the notation of [13] or [21, Page 164]:
Notation 3.24
Let \(X\in \mathscr {S}_*\) and \(f:A\rightarrow B\) in \(\mathrm {CAlg}\). By the previous lemma, there is an equivalence \(I(X\otimes _A B)\simeq J(X\otimes _A B)\) in \(\mathrm {Mod}_B\) natural in X and B: we denote the common value by \(X{{\,\mathrm{\widetilde{\otimes }}\,}}_A B\).
From Proposition 3.20 and Lemma 3.23, we deduce:
Corollary 3.25
Let \(f:A\rightarrow B\) in \(\mathrm {CAlg}\) and \(X\in \mathscr {S}_*\). There is an equivalence of B-modules
$$\begin{aligned} I(X\odot _B (B\wedge _A B)) \simeq X{{\,\mathrm{\widetilde{\otimes }}\,}}_A B \end{aligned}$$
natural in X.
We can now recast (3.12) in a different form. Define arrows \(e_n': S^n{{\,\mathrm{\widetilde{\otimes }}\,}}_A B \rightarrow \Omega (S^{n+1}{{\,\mathrm{\widetilde{\otimes }}\,}}_A B)\) as follows. Start by presenting \(S^{n+1}\in \mathscr {S}_*\) as the pushout of \(*\leftarrow S^n\rightarrow *\). Apply \(-\otimes _A B:\mathscr {S}_*\rightarrow \mathrm {CAlg}_{{{B}/ / {B}}}\) to it, and then apply I. Now \(e_n'\) is the universal pullback map \(S^n{{\,\mathrm{\widetilde{\otimes }}\,}}_A B \rightarrow \Omega (S^{n+1}{{\,\mathrm{\widetilde{\otimes }}\,}}_A B)\).
Proposition 3.26
Let \(f:A\rightarrow B\) in \(\mathrm {CAlg}\). There is an equivalence of B-modules
where \(\Omega \) denotes the loop functor in \(\mathrm {Mod}_B\).
Proof
We use the characterization \(L_{B/A}\simeq (P_1I)(B\wedge _A B)\) from (3.11). Consider the equivalence (3.12) with \(C=B\wedge _A B\). By Remark 3.14 and Corollary 3.25, we have
$$\begin{aligned} I(\Sigma ^n(B\wedge _A B)) \simeq I(S^n\odot _B (B\wedge _A B)) \simeq S^n{{\,\mathrm{\widetilde{\otimes }}\,}}_A B. \end{aligned}$$
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. \(\square \)
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 \({{\,\mathrm{\widetilde{\otimes }}\,}}\) construction and the \(E_k\)-cotangent complex mentioned at the beginning of the subsection.
Remark 3.28
Let B be an \(E_\infty \)-A-algebra. Forgetting structure, we may consider B as an \(E_k\)-A-algebra, for all \(k\ge 0\), and thus we may form its \(E_k\)-cotangent complex \(L^{(k)}_{B/A}\) [16, 7.3.5]. Lurie [16, 7.3.5.1/3] proved that for each \(k\ge 1\) there is a fiber sequence of B-modules
where \(*:S^{k-1}\rightarrow *\) denotes the unique map. Since \(*\otimes \mathrm {id}:S^{k-1}\otimes _A B\rightarrow B\) is the augmentation of \(S^{k-1}\otimes _A B\) and loops commute with pullbacks, this identifies \(L_{B/A}^{(k)}\) with \(\Omega ^{k-1}(S^{k-1}{{\,\mathrm{\widetilde{\otimes }}\,}}_A B)\) (see also [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 \(\infty \)-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 \(B\in \mathrm {CAlg}\). We denote by
$$\begin{aligned}Q:\mathrm {NUCA}_B\rightarrow \mathrm {Mod}_B\end{aligned}$$
the indecomposables functor that takes N to the cofiber in \(\mathrm {Mod}_B\) of the multiplication, i.e. to the pushout
in \(\mathrm {Mod}_B\).
The functor Q is left adjoint to the functor which takes a module M and endows it with a zero multiplication map. More precisely:
Lemma 3.30
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(1)
The functor Q is left adjoint to a functor \(Z:\mathrm {Mod}_B\rightarrow \mathrm {NUCA}_B\) such that \(UZ\simeq \mathrm {id}_{\mathrm {Mod}_B}\), and for each \(M\in \mathrm {Mod}_B\) the multiplication map \(ZM\wedge _B ZM \rightarrow ZM\) is zero.
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(2)
\(Q\circ F\simeq \mathrm {id}_{\mathrm {Mod}_B}\), where \(F:\mathrm {Mod}_B\rightarrow \mathrm {NUCA}_B\) is the free functor.
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(3)
There exists a unique functor \(Z:\mathrm {Mod}_B\rightarrow \mathrm {NUCA}_B\) such that \(UZ\simeq \mathrm {id}_{\mathrm {Mod}_B}\), up to equivalence.
Here \(U:\mathrm {NUCA}_B\rightarrow \mathrm {Mod}_B\) denotes the forgetful functor.
Proof
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(1)
We will use a criterion for adjointness from [15, 5.2.4.2]: Q admits a right adjoint if and only if for every \(M\in \mathrm {Mod}_B\), the comma \(\infty \)-category \((Q\downarrow M)\), defined as the pullback
has a terminal object. Fix \(M\in \mathrm {Mod}_B\). It suffices to see that there exists a B-nuca ZM such that \(M\simeq QZM\), for then \(QZM\simeq M\xrightarrow {\mathrm {id}} M\) is a terminal object of \((Q\downarrow M)\). To see this, first recall from Remark 3.6 that the augmentation ideal functor \(I:\mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {Mod}_B\) factors via an equivalence \(I_0:\mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {NUCA}_B\) followed by the forgetful functor. Now consider the trivial square-zero extension \(B\oplus M\) [16, 7.3.4.16]. This is an augmented \(E_\infty \)-B-algebra such that the multiplication map of \(I_0(B\oplus M)\) is zero. Define ZM to be \(I_0(B\oplus M)\), so clearly \(UZM\simeq M\). By definition, QZM is the cofiber in \(\mathrm {Mod}_B\) of the multiplication map \(ZM \wedge _B ZM\rightarrow ZM\) which is zero, so \(QZM\simeq M\) as required.
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(2)
\(Q\circ F\) is the left adjoint to \(U\circ Z\simeq \mathrm {id}_{\mathrm {Mod}_B}\), so it is equivalent to the identity.
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(3)
The existence of such a Z has just been proven. Now suppose we have a functor \(Z':\mathrm {Mod}_B\rightarrow \mathrm {NUCA}_B\) such that \(UZ'\simeq \mathrm {id}_{\mathrm {Mod}_B}\). Let \(M,M'\in \mathrm {Mod}_B\). We have natural equivalences of spaces
$$\begin{aligned} \mathrm {Map}_{\mathrm {NUCA}_B}(FM',Z'M)\simeq \mathrm {Map}_{\mathrm {Mod}_B}(M',UZ'M) \simeq \mathrm {Map}_{\mathrm {Mod}_B}(QFM',M). \end{aligned}$$
Let \(N\in \mathrm {NUCA}_B\). Note that the free-forgetful adjunction (F, U) is monadic by [16, 4.7.3.5], since \(\mathrm {NUCA}_B\) is an \(\infty \)-category of algebras over an \(\infty \)-operad in \(\mathrm {Mod}_B\) [16, 5.4.4.1] so the forgetful functor is conservative and preserves geometric realizations of simplicial objects [16, 3.2.2.6/3.2.3.2]. Therefore, by [16, 4.7.3.14/15], there exists a simplicial object \(N_\bullet \) in \(\mathrm {NUCA}_B\) which depends functorially on N and satisfies that \(\mathrm {colim}(N_\bullet )\simeq N\), \(N_\bullet \) is given by free nucas; moreover, \(\mathrm {colim}(QN_\bullet )\simeq QN\) by [16, 4.7.2.4]. Using the above, one obtains a natural equivalence of spaces
$$\begin{aligned} \mathrm {Map}_{\mathrm {NUCA}_B}(N,Z'M)\simeq \mathrm {Map}_{\mathrm {Mod}_B}(QN,M). \end{aligned}$$
By Remark 3.17, this proves that \(Z'\) is a right adjoint to Q, but then by uniqueness of adjoints [15, 5.2.6.2], we deduce that \(Z'\simeq Z\). \(\square \)
Remark 3.31
Let K denote the composition
$$\begin{aligned} \mathrm {Mod}_B\xrightarrow {\sim } \mathrm {Sp}(\mathrm {CAlg}_{{{B}/ / {B}}})\xrightarrow {\Omega ^\infty } \mathrm {CAlg}_{{{A}/ / {B}}}; \end{aligned}$$
here, the first arrow is an inverse to \(\Omega ^\infty \circ \partial I:\mathrm {Sp}(\mathrm {CAlg}_{{{B}/ / {B}}}) \xrightarrow {\sim }\mathrm {Mod}_B\). Note that K(M) is the trivial square-zero extension \(B\oplus M\) [16, 7.3.4.16]. By definition of K and \(L_{B/A}\), we get the following equivalence for every B-module M,
$$\begin{aligned} \mathrm {Map}_{\mathrm {Mod}_B}(L_{B/A},M) \simeq \mathrm {Map}_{\mathrm {CAlg}_{{{A}/ / {B}}}}(B,B\oplus M). \end{aligned}$$
These spaces can be interpreted as the spaces of A-linear derivations from B into M [16, 7.3].
Now, observe that K is equivalent to the composition
$$\begin{aligned}\mathrm {Mod}_B \xrightarrow {Z} \mathrm {NUCA}_B \xrightarrow {B\vee -} \mathrm {CAlg}_{{{B}/ / {B}}}\rightarrow \mathrm {CAlg}_{{{A}/ / {B}}},\end{aligned}$$
the last functor being a forgetful functor. Indeed, since \(B\vee -\) is an equivalence with inverse given by \(I_0\), we may equivalently verify that \(I_0\circ K \simeq Z\). But this follows from Part (2) of the previous lemma.
In the following theorem, we prove how to get the cotangent complex via indecomposables. A model categorical version of this result can be found in [8, Theorem 4].
Theorem 3.32
Let \(B\in \mathrm {CAlg}\). The following diagram commutes:
Proof
Recall from Sect. 2.(6) that \(P_1I\), the excisive approximation to I, is equivalent to \(\Omega ^\infty \circ \partial I \circ \Sigma ^\infty \). Thus, we have to prove that \(Q\circ I_0\) is equivalent to \(P_1I\). Recall that \(I\simeq U \circ I_0\). By [16, 6.1.1.30], we have \(P_1I \simeq P_1U \circ I_0\). We will now prove that \(P_1U \simeq Q\), finishing the proof.
By Sect. 2.(6), \(P_1U\) is equivalent to \(\Omega ^\infty _{\mathrm {Mod}_B} \circ \partial U \circ \Sigma ^\infty _{\mathrm {NUCA}_B}\). Note that Q is excisive, since it preserves pushouts and \(\mathrm {Mod}_B\) is stable, so \(Q\simeq P_1Q\). Therefore, to prove that \(P_1U\simeq Q\) it suffices to prove that \(\partial U\simeq \partial Q\), by Sect. 2.(6) once more.
We have the free–forgetful adjunction
, and taking derivatives gives an adjunction
by [16, 6.2.2.15]. By Lemma 3.30.(2), \(Q\circ F \simeq \mathrm {id}_{\mathrm {Mod}_B}\), so by [16, 6.2.1.4/24] we get \(\partial Q \circ \partial F \simeq \mathrm {id}_{\mathrm {Sp}(\mathrm {Mod}_B)}\). To prove that \(\partial Q\simeq \partial U\), it suffices to prove that \(\partial U\) is an equivalence. Indeed, if it is, then \(\partial F \circ \partial U\simeq \mathrm {id}_{\mathrm {Sp}(\mathrm {NUCA}_B)}\), so \(\partial Q \simeq \partial Q \circ \partial F \circ \partial U \simeq \partial U\).
To prove that \(\partial U\) is an equivalence, we proceed similarly as in [16, Proof of 7.3.4.7]: namely, since U is a monadic functor, then by [16, 6.2.2.17] it suffices to prove that the unit \(\mathrm {id}_{\mathrm {Mod}_B} \Rightarrow U\circ F\) induces an equivalence of derivatives. The proof of this is very similar to that of [16, 7.3.4.10], only simpler.
Note that \(U\circ F:\mathrm {Mod}_B\rightarrow \mathrm {Mod}_B\) is given on objects by \(M\mapsto M\vee \bigvee _{n\ge 2} (M^{\wedge _B n})_{\Sigma _n}\) and the unit of the (F, U) adjunction includes M into the separate M factor, so by [16, 7.3.4.8] it suffices to see that the derivative of the functor \(\mathrm {Sym}^n:\mathrm {Mod}_B\rightarrow \mathrm {Mod}_B, M\mapsto (M^{\wedge _B n})_{\Sigma _n}\) is nullhomotopic for \(n\ge 2\).
To see this, note that \(\mathrm {Sym}^n\) is the composition
where the first functor is the diagonal functor and the second functor takes \((M_1,\dots ,M_n)\) to \(M_1\wedge _B \cdots \wedge _B M_n\) together with the action by \(\Sigma _n\) which permutes the factors. By [16, 7.3.4.8] the derivative operator \(\partial :\mathrm {Fun}_*(\mathrm {Mod}_B,\mathrm {Mod}_B)\rightarrow \mathrm {Exc}(\mathrm {Mod}_B,\mathrm {Mod}_B)\) preserves colimits, so the derivative of \(\mathrm {Sym}^n\) is the colimit of a functor \(B\Sigma _n \rightarrow \mathrm {Exc}(\mathrm {Mod}_B,\mathrm {Mod}_B)\) with value \(\partial \left( (-)^{\wedge _B n}:\mathrm {Mod}_B\rightarrow \mathrm {Mod}_B\right) \). Therefore, it suffices to see that the functor \((-)^{\wedge _B n}:\mathrm {Mod}_B\rightarrow \mathrm {Mod}_B\), \(n\ge 2\) has nullhomotopic derivative.
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.Footnote 5 By Sect. 2.(6), its derivative is nullhomotopic as well, since \(\mathrm {Mod}_B\) is stable. \(\square \)
Remark 3.33
The key aspect of the previous proof is the fact that \(\partial (U \circ F) \simeq \mathrm {id}_{\mathrm {Sp}(\mathrm {Mod}_B)}\), which is proven in the last paragraph. Notice this is equivalent to
$$\begin{aligned} P_1(U \circ F) \simeq \mathrm {id}_{\mathrm {Mod}_B}, \end{aligned}$$
as \(P_1(U \circ F) \simeq \Sigma ^\infty \circ \partial (U \circ F) \circ \Omega ^\infty \) by Sect. 2.(6), and \(\Sigma ^\infty , \Omega ^\infty \) are equivalences since \(\mathrm {Mod}_B\) is stable. Using the analogy of Goodwillie calculus with classical calculus, we can gain some intuition for this result. Under this analogy, functors correspond to smooth functions of the real line, so the functor \(U \circ F\) which maps \(M\in \mathrm {Mod}_B\) to \(\bigvee _{n\ge 1} (M^{\wedge _B n})_{\Sigma _n}\) corresponds to the power series \(f(x) = \sum _{n=1}^\infty x^n\). The linear approximation at 0 of this function is the identity map \(x\mapsto x\). Continuing with the analogy, linear approximations of functions correspond to 1-excisive approximations of functors, which provides some intuition for the equivalence \(P_1(U \circ F) \simeq \mathrm {id}_{\mathrm {Mod}_B}\).
From Theorem 3.32 we immediately deduce:
Corollary 3.34
Let \(f:A\rightarrow B\) be a map of \(E_\infty \)-ring spectra. There is an equivalence of B-modules
$$\begin{aligned}L_{B/A}\simeq QI_0(B\wedge _A B).\end{aligned}$$
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.