Deformations of E ∞ -groups of units and logarithmic derivatives of E ∞ -rings
DDEFORMATIONS OF E ∞ -GROUPS OF UNITS ANDLOGARITHMIC DERIVATIVES OF E ∞ -RINGS STEFANO ARIOTTA
Abstract.
We extend a classical fact about deformations of groupsof units of commutative rings to E ∞ -ring spectra, and we use this re-sult to provide a map of spectra generalizing the ordinary logarithmicderivative induced by an R -module derivation. Contents
Introduction 11. Groups of units and square-zero extensions 31.1. Definitions and basic facts 31.2. Square-zero extensions 101.3. Groups of units of square-zero extensions, the setup 151.4. Groups of units of square-zero extensions, top-down 181.5. Groups of units of square-zero extensions, bottom-up 182. Logarithmic derivatives of E ∞ -rings 20Appendix A. The classical result 25Appendix B. Fundamentals of higher commutative algebra 26B.1. Symmetric monoidal ∞ -categories 26B.2. Modules over commutative algebra objects 31References 32 Introduction
Stable homotopy theory provides, among other things, a framework todevelop a homotopy coherent version of ordinary commutative algebra. In-stances of this generalization are by now abundant in the literature. Theaim of this paper is to extend to this context the notion of logarithmicderivatives, by means of a generalization of the following classical result(see Proposition A.3 for a more detailed statement, and a proof).
Proposition
Given a square-zero extension of commutative rings (cid:101) R → R with kernel I , there exists an induced short exact sequence of Abelian groups0 −→ I −→ GL (cid:101) R −→ GL R −→ . a r X i v : . [ m a t h . A T ] S e p ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 2 The main motivation for the definition of homotopy coherent logarithmicderivatives is the investigation of lifts of Adams operations on (suitablelocalizations of) K-theory to morphisms of spectra, developing on currentwork in progress by Barwick-Glasman-Mathew-Nikolaus.To work with homotopy theoretic tools, we will use the language of ∞ -categories, as developed in [Lur09], and we will set our discussion of stablehomotopy theoretic ideas in the framework developed, using this language,in [Lur17].To guide the intuition in the homotopy coherent setting, it is useful tothink of the ∞ -category of spaces as playing the role pertaining to thecategory of sets in the ordinary context. We can express the analogy withordinary commutative algebra by thinking of the ∞ -category of spectraas the analogue of the Abelian category of Abelian groups and the derivedcategory of the integers at once (and more generally of stable ∞ -categories asthe analogue of both Abelian categories and derived categories). Along theselines, E ∞ -rings become the counterparts of ordinary commutative rings.With these ideas in mind, our goal will be to prove the following gener-alization of the above Proposition (see Theorem 1.30; the reasons for theconnectivity conditions will be clearified in Remark 1.31). Theorem
Let R be a connective E ∞ -ring, and let (cid:101) R → R be a square-zero extension by a connective R -module M . Then, there exists a co/fibersequence M → gl (cid:101) R → gl R in the ∞ -category Sp.By virtue of the above theorem, we will be able to define a map of spectralog ∂ : gl R → M from any connective E ∞ -ring R and any derivation (asdefined in [Lur17, 7.4.1] and reviewed in Section 1.2) ∂ of R into an R -module M .In Section 1, we set up what we need in order to state the theorem, thenwe give two alternative proofs of it, one recovering the ordinary propositionas a particular case, the other leveraging the homotopy coherent statementfrom the ordinary one. In Section 2, we define our homotopy coherent loga-rithmic derivative, and show how it generalizes the ordinary one. AppendixA, contains a proof of the ordinary result. Finally, in Appendix B, we re-call a few results about symmetric monoidal ∞ -categories and modules overcommutative algebra objects. Prerequisites.
We assume the reader is familiar with the language of ∞ -categories, as developed in [Lur09]. Notably, we make free use of the AdjointFunctor Theorem (see [Lur09, 5.5.2.9]). Even if we will recall some defini-tions, we will assume that the reader is familiar with the theory of stable ∞ -categories and of ∞ -operads as developed in [Lur17], and in particularwith the ∞ -category of spectra and its symmetric monoidal structure given ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 3 by the smash product. Some of the notations used in this paper differ fromthe ones in [Lur17]. However, we use the same notations and terminologyof [Lur17] for all the concepts we do not explicitly recall or introduce here. Relation to previous work.
A result essentially equivalent to Theorem1.30 appeared, using rigid models to deal with homotopy coherent struc-tures, as [Rog09, Lemma 11.2], and the proof sketched in [Rog09, Remark11.3] is strictly related to the proof we present in Section 1.4.
Aknowledgements.
Most of the results of this paper were obtained aspart of the author’s M.Sc. Thesis at the University of Bonn in 2018. I amextremely grateful to my advisor, Thomas Nikolaus, for proposing me thistopic and for his thorough support and his invaluable advice. I would liketo thank Jack Davies, Simone Fabbrizzi, Giulio Lo Monaco, Fosco Loregian,Riccardo Pedrotti and Mauro Porta for the insightful discussions I had thepleasure to be involved in while writing this article.1.
Groups of units and square-zero extensions
We assume the reader is familiar with the theories of stable ∞ -categoriesand of ∞ -operads, as developed in [Lur17]. For the convenience of thereader, and to fix notations, we recall some definitions and some resultsabout symmetric monoidal ∞ -categories and modules over (homotopy co-herent) commutative algebra objects in Appendix B, and we refer to [Lur17]for all the concepts there undefined.We focus our attention to the presentable symmetric monoidal ∞ -cate-gories of spaces and spectra. In Section 1.1, we begin by defining the objectswe want to investigate, namely groups of units and square-zero extensionsof E ∞ -rings, and by establishing some of their basic properties. Then, inSection 1.3, we begin our proof of Theorem 1.30. In Section 1.4 and 1.5we provide two alternative proofs, one at the spectral level, and one at thespace level.1.1. Definitions and basic facts.
Most of the material presented in thissection, is an adaptation of definitions and results presented using rigidmodels for the categories of spectra in [ABG +
14] and [Rez06]. In particular,our treatment of groups of units and group E ∞ -rings is inspired by thepresentation given in [ABG + ∞ -categoricalsetting. Also, we recollect various facts regarding the properties of groupsof units of E ∞ -rings following closely [Rez06], working out explicitly somedetails left to the reader therein. Finally, we recall some definitions andbasic facts about square-zero extensions of E ∞ -rings from [Lur17].We will reserve the notations CMon, CGrp and CAlg for the ∞ -categoriesCMon( S ), CGrp( S ) and CAlg(Sp), respectively (see Remark B.13). ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 4 Proposition 1.1
The symmetric monoidal adjunction Σ ∞ + (cid:97) Ω ∞ , restrictsto an adjunction S [ − ] : CMon (cid:29) CAlg : Ω ∞ m such that the diagram CMon CAlg S Sp S [ − ]Σ ∞ + (where the vertical arrows are the forgetful functors) commutes. Proof.
Recall that CMon (cid:39)
Alg E ⊗∞ ( S ) ⊂ Fun E ⊗∞ ( E ⊗∞ , S × ). The functor S [ − ] is obtained by postcomposition with the symmetric monoidal functor S × → Sp ⊗ induced by Σ ∞ + , and Ω ∞ m is obtained analogously.As explained in [RV15, RV17], to give an adjunciton between ∞ -cate-gories is equivalent to give an adjunction in the underlying 2-category ofquasi-categories. Given unit and counit transformations for the adjunctionΣ ∞ + (cid:97) Ω ∞ , we obtain unit and counit transformations for S [ − ] and Ω ∞ m byprecomposition with (the underlying 1-cells of) sections of the symmetricmonoidal structure maps, thus satisfying the triangle identities automati-cally. (cid:3) Definition 1.2
Given an E ∞ -ring R , we call Ω ∞ m R its underlying multiplica-tive E ∞ -space . Given an E ∞ -space X , we call S [ X ] its monoid E ∞ -ring .A few words on notation and terminology are overdue. To see the reasonsfor the analogy with the ordinary case, consider an E ∞ -ring R . By defini-tion, R is a commutative algebra object in the ∞ -category of spectra. Thefunctor Ω ∞ m acts by “forgetting” the additive structure of R (falling backfrom spectra to spaces), but still remembering its multiplicative structure,as it takes values in the ∞ -category of commutative algebra objects in S .The analogy is particularly clear if we restrict to connective E ∞ -rings. ByRemark B.21, the ∞ -categories Sp cn and CGrp are equivalent, hence the ∞ -category of connective E ∞ -rings CAlg cn is equivalent to the ∞ -categoryCAlg(CGrp). As the restriction of Ω ∞ to Sp cn corresponds to the forget-ful functor CGrp → S under this equivalence, the functor Ω ∞ m restricts toCAlg(CGrp) → CAlg( S ) (cid:39) CMon.
Proposition 1.3
The forgetful functor CGrp → CMon admits a rightadjoint, denoted ( − ) × : CMon → CGrp . Proof.
This is a direct consequence of [Lur17, 5.2.6.9] and the Adjoint Func-tor Theorem. (cid:3)
Remark 1.4
The functor ( − ) × can be explicitly described as the functorsending an E ∞ -space M to its connected components which are invertible ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 5 in the commutative monoid π M (see Remark B.10), and the counit of theadjunction as the direct summands inclusion. To see this, let us momentarilydenote by M u those connected components of M which are invertible in π M , and let us denote by ι : M u → M the direct summands inclusion.Clearly, given any E ∞ -group G , postcomposition with ι determines a mapMap CGrp ( G, M u ) ι ∗ −→ Map
CMon ( G, M ) . On the other hand, any element in Map
CMon ( G, M ) induces a morphismof ordinary monoids π G → π M , which, since π G is a group, factorsthrough ( π M ) × . As a consequence, the actual morphism G → M westarted from has to factor through ι . A straightforward check shows thatthis factorization gives a homotopy inverse to ι ∗ .Using the notation given in Appendix B, we think of ( − ) gp as groupcompletion and of ( − ) × as the maximal subgroup of an E ∞ -space. Notation 1.5
We denote the composite right adjoint ( − ) × ◦ Ω ∞ m byGL : CAlg → CGrpand its left adjoint, using a notation analogous to the common one for theordinary case, again by S [ − ]; we denote the composite functor B ∞ GL by gl : CAlg → Sp(see Proposition B.17 for the definition of B ∞ ). Given an E ∞ -ring R , wewill use the term E ∞ -group of units of R to refer to GL R ; we will call gl R the units-group spectrum of R . We will sometimes just use the term group of units to refer to either of the two. Notation 1.6
Let X be a space, and let R be an E ∞ -ring. We will use thenotation H • ( X ; gl R )to denote the cohomology groups π Map Sp (cid:0) Σ ∞ + X, gl R [ • ] (cid:1) . Given any pointed space X , we will denote by (cid:101) H • ( X ; gl R )the reduced cohomology groups π Map Sp (Σ ∞ X, gl R [ • ]) . Remark 1.7
The counit of the adjunction given by Remark 1.4 gives anatural transformation ι : GL ⇒ Ω ∞ m . It follows from the same remark ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 6 that, given any E ∞ -ring R , the map ι R fits into a Cartesian squareGL R Ω ∞ m R ( π R ) × π R ι R in the ∞ -category CMon (where both ( π R ) × and π R are considered asdiscrete E ∞ -spaces) and thus in the ∞ -category S . In particular, as all thepath components of a grouplike H-space are homotopy equivalent, gl R isa connective spectrum having as homotopy groups: π n ( gl R ) = (cid:40) π ( R ) × for n = 0; π n ( R ) for n ≥ Remark 1.8
As a consequence of Remark 1.7, we have that for any E ∞ -ring R , the following isomorphism of groups holds H ( X ; gl R ) (cid:39) (cid:0) R X (cid:1) × . In fact, as the (multiplicative) commutative monoid structure of R X (cid:39) π Map S ( X, Ω ∞ R )is determined by the commutative monoid object structure of Ω ∞ R in thehomotopy category ho S , the maps X → Ω ∞ R factoring through GL R correspond exactly to the invertible elements in R X . Remark 1.9
Given any E ∞ -monoid X , the E ∞ -ring S [ X ] comes with anaugmentation map S [ X ] → S that is natural and compatible with all therelevant adjunctions (see Remark 1.11 for an explicit description of thismap).More precisely, both functors denoted S [ − ] factor through the ∞ -cate-gory CAlg / S of augmented E ∞ -rings . In fact, since S [0] (cid:39) S , we have aninduced functor CGrp (cid:39) CGrp / → CAlg / S whose composition with the forgetful functor CAlg / S → CAlg is precisely S [ − ]. By [Lur09, 1.2.13.8], the forgetful functor CAlg / S → CAlg preservescolimits; hence, by the Adjoint Functor Theorem it admits a right adjoint.As a consequence, we have that the adjunction S [ − ] (cid:97) Ω ∞ m (resp. S [ − ] (cid:97) GL ) factors as a composite adjunctionCMon (cid:29) CAlg / S (cid:29) CAlg(resp. CGrp (cid:29)
CAlg / S (cid:29) CAlg).
Remark 1.10
Given any connected pointed space X , the terminal mor-phism X → pt admits as a section the basepoint inclusion pt → X . Byapplying the free basepoint functor ( − ) + introduced in Proposition B.17 to ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 7 the basepoint inclusion, we obtain a pointed map S → X + , which in turnfits into a cofiber sequence of pointed spaces S → X + → X where the last map is the counit of the free-forgetful adjunction S (cid:29) S ∗ . Byapplying the functor Σ ∞ to the above sequence, we get a co/fiber sequencein Sp: S → Σ ∞ + X → Σ ∞ X. Since the first map admits a retraction (the image under Σ ∞ + of X → pt),the co/fiber splits and Σ ∞ + X (cid:39) Σ ∞ X ⊕ S . Following [Rez06], we will denote by γ X : Σ ∞ X → Σ ∞ + X the section (well-defined up to homotopy) of the projection Σ ∞ + X → Σ ∞ X . Remark 1.11
The above remark lets us give an explicit description of theaugmentation map of a monoid or group E ∞ -ring. Let X be an E ∞ -space,and let a : S [ X ] → S be its monoid E ∞ -ring, together with the augmentationmap given by Remark 1.9. Looking at the map underlying a in Sp, we havethat it is obtained by applying the functor Σ ∞ + to the terminal morphism X → pt, hence it is given by the morphism S ⊕ Σ ∞ X → S ⊕ (cid:39) S acting as the identity on S , and as the zero morphism elsewhere, informallywritten as (cid:18) id S
00 0 (cid:19) . Notation 1.12
Let A and B be spaces, let X and Y be pointed spaces andlet H and K be spectra. We will sometimes use the notations[ A, B ], [
X, Y ] ∗ and [ H, K ]to denote π Map S ( A, B ), π Map S ∗ ( X, Y ) and π Map Sp ( H, K )respectively.
Remark 1.13
Let E be a spectrum. The splitting discussed in Remark 1.10induces the usual direct sum decomposition for the unreduced cohomologyof a pointed space, i.e. for every n ∈ Z , it induces an isomorphism E n ( X ) (cid:39) (cid:104) Σ ∞ + X [ − n ] , E (cid:105) (cid:39) (cid:104) Σ ∞ X [ − n ] , E (cid:105) ⊕ (cid:104) S [ − n ] , E (cid:105) (cid:39) (cid:101) E n ( X ) ⊕ π − n E. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 8 In particular, this, together with the isomorphism H ( X ; gl R ) (cid:39) (cid:0) R X (cid:1) × implies that H ( X ; gl R ) (cid:39) (cid:16) (cid:101) R X ⊕ π R (cid:17) × (cid:39) [Σ ∞ X, gl R ] ⊕ [ S , gl R ] (cid:39) (cid:101) H ( X ; gl R ) ⊕ ( π R ) × . Thence, the retraction Σ ∞ + X → Σ ∞ X induces an isomorphism (cid:101) H ( X ; gl R ) (cid:39) (cid:16) (cid:101) R X + 1 π R (cid:17) × ⊂ (cid:16) (cid:101) R X ⊕ π R (cid:17) × . Definition 1.14
Let ( X, ∗ ) be a pointed space and let E be a spectrum.Given an unpointed map f : X → Ω ∞ E , we define the basepoint shift Sh ( f )of f to be the composite pointed map X (cid:39) X × pt id ×∗ −−−→ X × X f × (inv ◦ f ) −−−−−−→ Ω ∞ E ⊕ Ω ∞ E µ −→ Ω ∞ E where µ denotes the multiplication map on Ω ∞ E , and inv denotes its in-version map (both given equivalently by the fact that Ω ∞ E is the spaceunderlying the additive E ∞ -group Ω ∞ a E , or by Remark B.22). Informally,we have that Sh ( f )( x ) = f ( x ) − f ( ∗ ) . Remark 1.15
The assignment Sh : f (cid:55)→ Sh ( f ) determines a function[ X, Ω ∞ E ] [ X, Ω ∞ E ] ∗ (cid:101) E X ⊕ π E (cid:101) E X Sh (cid:39) (cid:39) corresponding to the projection of unreduced cohomology to the reducedcohomology direct summand. That is, the map E X → (cid:101) E X induced by Sh corresponds to the map[Σ ∞ + X, E ] ( γ X ) ∗ −−−→ [Σ ∞ X, E ]induced by precomposition with the inclusion γ X introduced in Remark1.10. In other words, given any map f ∈ [ X, Ω ∞ E ], the map Sh ( f ) is theadjoint of the composite map f (cid:91) ◦ γ X ∈ [Σ ∞ X, E ], f (cid:91) being the adjointmap of f . In particular, Sh preserves whatever additive or multiplicativestructure the spectrum E may induce on the set [Σ ∞ + X, E ]. see Notation B.20 ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 9 Let R be an E ∞ -ring. The map ι R : GL R → Ω ∞ m R given by Remark 1.7 is of course unpointed. Given any pointed space X ,we have an induced morphism( ι R ) ∗ : [ X, GL R ] ∗ → [ X, Ω ∞ R ]and, by postcomposition with Sh , we get a morphism Sh ◦ ( ι R ) ∗ : [ X, GL R ] ∗ → [ X, Ω ∞ R ] ∗ . We want to prove that for X = S k and k ≥
1, the above map is an isomor-phism of Abelian groups.
Proposition 1.16
Let R be an E ∞ -ring. Given any k ≥
1, the map Sh ◦ ( ι R ) ∗ induces an isomorphism[ S k , GL R ] ∗ (cid:39) [ S k , Ω ∞ R ] ∗ . Proof. [ S k , GL R ] ∗ consists of homotopy classes of maps S k → GL R send-ing S k to the path component 1 π R ∈ ( π R ) × . By postcomposing with ι R : GL R → Ω ∞ R (which is an unpointed map), we get to the subset( ι R ) ∗ [ S k , GL R ] ∗ ⊂ [ S k , Ω ∞ R ]of maps sending S k to the path component 1 π R ∈ π R that are pointed at1 π R ; thus, compatibly with Remark 1.13, ι R realizes the inclusion of thesubgroup (cid:16) (cid:101) R S k + 1 π R (cid:17) × in the multiplicative monoid of the ring R S k . If we now compose with themap Sh intruduced in Remark 1.15, we get to pointed maps [ S k , Ω ∞ R ] ∗ , orequivalently, to the group π k R (cid:39) (cid:101) R S k ⊂ ( R S k , · R ) Our claim is that themap [ S k , GL R ] ∗ Sh ◦ ( ι R ) ∗ −−−−−→ [ S k , Ω ∞ R ] ∗ (1)is a group isomorphism. By construction, ( ι R ) ∗ is a group homomorphisminto the multiplicative structure of its target, whereas by Remark 1.15, Sh induces a homomorphism between the multiplicative structures of itssource and target; hence it is sufficient to provide an inverse (as a set-map)to Sh ◦ ( ι R ) ∗ .As (1) is just a change of connected component, from 1 π R to 0 π R , it issufficient to realize the inverse change. To this end, let us consider the map[ S k , Ω ∞ R ] ∗ − + −−−→ [ S k , Ω ∞ R ]where denotes the map S k → S π R −−−→ Ω ∞ R , and the sum comes from thecogroup object structure on S k . Under − + , all the maps in [ S k , Ω ∞ R ] ∗ are shifted to the path component of 1 π R ∈ π R . By Remark 1.7, each ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 10 homotopy class in [ S k , Ω ∞ R ] ∗ + determines a (unique, up to homotopy)map S k → GL R , sending S k to the path component of 1 π R ∈ GL π R .In particular, the universal property of the pullback induces a map[ S k , Ω ∞ R ] ∗ −→ [ S k , Ω ∞ R ] ∗ + −→ [ S k , GL R ] ∗ . (2)As this composition has the effect of shifting from the connected componentof 0 π R to that of 1 π R , this is an inverse for (1). (cid:3) Remark 1.17
It follows from Remark 1.13 that ( (cid:101) R S k + 1 π R ) × (cid:39) (cid:101) R S k .The proof of Proposition 1.16 shows that, moreover, an isomorphism is givenby x + 1 (cid:55)→ x .1.2. Square-zero extensions.
We will briefly recall the definitions of (triv-ial and general) square-zero extensions of E ∞ -rings, referring the readerto [Lur17, Chapter 7] for a detailed treatment. Remark 1.18
Let A be an E ∞ -ring. In [Lur17, 7.3.4.15] a functor A ⊕ − : Mod A → CAlg /A is constructed, sending an A -module M to the trivial square-zero extensionof A by M . As explained in [Lur17, 7.3.4.16] the notation is motivated bythe fact that it sends each object M ∈ Mod A to a commutative E ∞ -ring over A whose underlying spectrum is equivalent to A ⊕ M in Sp. The algebrastructure on A ⊕ M is “square-zero” in the homotopy category of CAlg, asthe following facts hold:(1) The unit map S → A ⊕ M is homotopic to the composition of theunit map S → A with the inclusion A → A ⊕ M .(2) The multiplication( A ⊗ A ) ⊕ ( A ⊗ M ) ⊕ ( M ⊗ A ) ⊕ ( M ⊗ M ) (cid:39) ( A ⊕ M ) ⊗ ( A ⊕ M ) → A ⊕ M is given as follows: • On A ⊗ A , it is homotopic to the composition of the multipli-cation of A with the inclusion A → A ⊕ M . • On A ⊗ M and M ⊗ A , it is given by composition of the actionof A on M with the inclusion M → A ⊕ M . • On M ⊗ M it is nullhomotopic. Remark 1.19
The functor A ⊕ − introduced above admits a left adjoint L A : CAlg /A → Mod A whose value on A (with the identity as structure map) is denoted L A andcalled the cotangent complex of A .In order to describe the behavior of trivial square-zero extensions underrestriction of scalars, we need to recall the following result. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 11 Proposition 1.20 [Lur17, 7.3.4.14] Let A be an E ∞ -ring. There is acanonical equivalence of ∞ -categoriesSp(CAlg /A ) (cid:39) Mod A . Proof.
It follows from [Lur17, 1.4.2.18] that Sp(CAlg /A ) (cid:39) Sp(CAlg
A//A ).Moreover, by [Lur17, 3.4.1.7], there exists an equivalence CAlg A/ (cid:39) CAlg(Mod A ).Combining the two, we get:Sp(CAlg /A ) (cid:39) Sp(CAlg(Mod A ) /A ) . The objects in CAlg(Mod A ) /A are, by definition, equipped with a structuremap with target A , in the ∞ -category Mod A . Hence, taking fibers of thestructure maps gives a functor CAlg(Mod A ) /A → Mod A , which, being leftexact, in turn induces a functorExc ∗ ( S fin ∗ , CAlg(Mod A ) /A ) Exc ∗ ( S fin ∗ , Mod A )Sp(CAlg(Mod A ) /A ) Sp(Mod A ) (cid:39) Mod A by pointwise composition (where the bottom right equivalence is due to thefact that Mod A is stable). To conclude, [Lur17, 7.3.4.7] shows that this lastfunctor is an equivalence. (cid:3) Remark 1.21
Let f : A → B be a morphism of E ∞ -rings. Then, there isan induced adjunction CAlg /A (cid:29) CAlg /B , where the left adjoint is givenby postcomposition, and the right adjoint is given by pullback along f . Letus denote by f ∗ : CAlg /B → CAlg /A the right adjoint functor. The functor f ∗ restricts to a functor F : CAlg B//B → CAlg
A//A by commutativity of the following diagram
A BR × B A RA B. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 12 The functor F , in turn, induces by pointwise composition the horizontalfunctors in the following diagramExc ∗ ( S fin ∗ , CAlg
B//B ) Exc ∗ ( S fin ∗ , CAlg
A//A )Sp(CAlg
B//B ) Sp(CAlg
A//A )Sp(CAlg(Mod B ) /B ) Sp(CAlg(Mod A ) /A )Exc ∗ ( S fin ∗ , CAlg(Mod B ) /B ) Exc ∗ ( S fin ∗ , CAlg(Mod A ) /A ) . (cid:39) (cid:39) By inspection (recall that in Proposition 1.20 the equivalence CAlg(Mod A ) /A (cid:39) Mod A was induced by taking fibers of the structure maps), this functor isequivalent to the restriction of scalars functor f ! : Mod B → Mod A . In partic-ular, this implies that, given any B -module M , the E ∞ -rings ( B ⊕ M ) × B A and A ⊕ f ! M are equivalent, and that the square A ⊕ f ! M B ⊕ MA B f is Cartesian in the ∞ -category CAlg. Definition 1.22
Let R be an E ∞ -ring, and let M be an R -module. Wedefine a derivation from R to M to be a map of R -modules L R → M. As it is clear, a derivation is equivalently determined (up to a contractiblespace of choices) by its adjoint map R → R ⊕ M (over R ). If η : L R → M is a derivation, we will denote its adjoint map by d η : R → R ⊕ M . Definition 1.23
Let R be an E ∞ -ring, and let η : L R → M be a derivationfrom R to an R -module M . Let ϕ : (cid:101) R → R be a morphism in CAlg. Wewill say that ϕ is a square-zero extension if there exists a Cartesian square (cid:101) R RR R ⊕ M ϕ d η d in the ∞ -category CAlg (where d is adjoint to the zero map L R → M ). Inthis case, we will also say that (cid:101) R is a square-zero extension of R by M [ − ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 13 Remark 1.24
Differently from the classical case, being square-zero is not aproperty of an extension, but an additional structure. Given a square-zeroextension f : (cid:101) R → R , in general, both the module M and the derivation η need not be uniquely determined, even up to equivalence. However, thishappens in the most common situations (see [Lur17, 7.4.1.26]).We conclude this section giving a characterization of the values of L S for monoid and group E ∞ -rings. This result will constitute a key step forthe first proof of Theorem 1.30. A version of Proposition 1.25 is provedin [BM05, 6.1] using rigid models for the categories of spectra. After settingup a few preliminary results, we present a purely ∞ -categorical proof of it. Proposition 1.25
The functor L S ◦ S [ − ] : CMon → Spis naturally equivalent to B ∞ ◦ ( − ) gp (see Appendix B). In particular, forany connective spectrum M , we have a natural equivalence L S ( S [Ω ∞ a M ]) (cid:39) M. In order to prove Proposition 1.25, we will first introduce the analogue ofthe symmetric algebra functor, sending a spectrum X to the free E ∞ -ringSym * X . The free-forgetful adjunction Sp (cid:29) CAlg thus obtained is com-patible with the free-forgetful adjunction S (cid:29) CMon given by PropositionB.17, in the sense made precise by Proposition 1.27. Moreover, the freefunctor Sym * will naturally be augmented, and in its augmented fashion,will constitute a right inverse functor for L S . Proposition 1.26
There exists a free functorSym * : Sp → CAlgleft adjoint to the forgetful functor of Remark B.9.
Proof.
This is an immediate consequence of [Lur17, 3.1.3.5]. (cid:3)
Proposition 1.27
Given a space X , there exists a natural equivalence S [ F + X ] (cid:39) Sym * Σ ∞ + X where F + is the functor defined in Proposition B.17. Proof.
Let us consider the following diagram S CMonSp CAlg . F + Σ ∞ + (cid:97) (cid:97) S [ − ] (cid:97) Sym * Ω ∞ (cid:97) Ω ∞ m ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 14 We can reformulate what we want to prove by saying that the square de-termined by the left adjoints commute. Since the square determined by theright adjoints clearly commutes, we are done. (cid:3)
Similarly to what happens for S [ − ] (see Remark 1.9), the free-forgetfuladjunction Sp (cid:29) CAlg factors through the ∞ -category CAlg / S as a com-posite adjunction Sp (cid:29) CAlg / S (cid:29) CAlg . Proposition 1.28 [Lur17, 7.3.4.5] There exists an adjunctionSym ∗ aug : Sp (cid:29) CAlg / S : (cid:73) with the following properties:(1) The functor Sym ∗ aug is given by compositionSp (cid:39) Sp / * −−−→ CAlg / S . (2) The functor (cid:73) is given by taking fibers of the structure maps inCAlg / S .(3) The composition Sp Sym ∗ aug −−−−→ CAlg / S (cid:73) −→ Spis equivalent to X (cid:55)→ (cid:81) n> Sym n ( X ) (see [Lur17, 3.1.3.9]).We will often abuse notation, and denote Sym ∗ aug just by Sym * . Proposition 1.29
The composition L S Sym * is naturally equivalent to theidentity functor. Proof.
Let X and Y be spectra. We have a chain of equivalencesMap Sp ( L S Sym * X, Y ) (cid:39) Map
CAlg / S (Sym * X, S ⊕ Y ) (cid:39) Map Sp ( X, (cid:73) ( S ⊕ Y )) (cid:39) Map Sp ( X, Y )from which we deduce that L S Sym * and id Sp represent the same functor inthe ∞ -category Sp, and are therefore naturally equivalent. (cid:3) We are now ready to prove Proposition 1.25.
Proof of Proposition 1.25.
By [GGN15, 4.9], we have a natural equivalenceFun L (CMon , Sp) ∼ −→ Sp ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 15 given by evaluation at F + (pt), the free E ∞ -space generated by one point.By what we have seen so far, we have that L S ( S [ F + (pt)]) (1 . (cid:39) L S (Sym * Σ ∞ + (pt)) (1 . (cid:39) Σ ∞ + (pt) ( B. (cid:39) B ∞ ( F + (pt)) gp hence L S ◦ S [ − ] and B ∞ ◦ ( − ) gp are naturally equivalent. The second partof the statement just follows from the equivalence between E ∞ -groups andconnective spectra (see Remark B.21) and the fact that ( − ) gp is the identityon E ∞ -groups, as it is left adjoint to the inclusion CGrp → CMon. (cid:3)
Groups of units of square-zero extensions, the setup.
Our goalin this and the following two sections is to give two proofs of the followinggeneralization of Proposition A.3 in our homotopy coherent setting.
Theorem 1.30
Let R be a connective E ∞ -ring, and let (cid:101) R → R be a square-zero extension by a connective R -module M . By applying gl to (cid:101) R → R ,we obtain a map of spectra ϕ : gl (cid:101) R → gl R. The fiber of ϕ is naturally equivalent to M in the ∞ -category Sp. Remark 1.31 As gl (introduced in Notation 1.5) lands, by definition,in the image of B ∞ (defined in Proposition B.17), upon its application,all nonconnective information is lost (see Remark B.21). Hence, we canrestrict ourselves to work with connective E ∞ -rings and connective moduleswithout loss of generality, and suitably replace the relevant objects withtheir connective covers when dealing with nonconnective ones.Our strategy for the proof will be the following: in this section, we willfirst show how to reduce the problem from general square-zero extensionsto trivial ones, and then how to reduce it further to trivial square-zeroextensions of the sphere spectrum S . In the following two sections, wewill show that the theorem indeed holds in this last case. In Section 1.4,we will do the last step in an exquisitely “higher algebraic” fashion, bygiving a proof entirely at the level of spectra, and in particular recoveringthe ordinary result as a particular case. In Section 1.5, we will give analternative proof substantially founded on the space level, extending thehomotopy coherent result from the ordinary one. Proposition 1.32
Let R be a connective E ∞ -ring, and let M be any con-nective R -module. Let gl ( R ⊕ M ) → gl R ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 16 be the map obtained by applying the functor gl to the trivial square-zeroextension R ⊕ M → R and let 1+ M denote the fiber of gl ( R ⊕ M ) → gl R .Then given any square-zero extension (cid:101) R → R of R by M , the fiber of theinduced map gl (cid:101) R → gl R is naturally equivalent to 1 + M . Proof.
First of all, we observe that gl ( R ⊕ M ) (cid:39) gl R ⊕ (1 + M ). Infact, since the map of E ∞ -rings R ⊕ M → R admits a section, the sameis true for its image under the right adjoint functor gl , and hence theco/fiber sequence 1 + M → gl ( R ⊕ M ) → gl R splits. In particular,1 + M (cid:39) coker ( gl R → gl ( R ⊕ M )), so that(1 + M )[1] (cid:39) M [1]) . By virtue of this canonical identification, we will unambiguously just write1 + M [1] . Let us now suppose that (cid:101) R → R is a square-zero extension of R by M .By definition, (cid:101) R sits in a Cartesian square (cid:101) R RR R ⊕ M [1]in the ∞ -category CAlg. Upon applying the functor gl , we obtain theco/Cartesian square gl (cid:101) R gl R gl R gl R ⊕ (1 + M [1])in the ∞ -category Sp.The result now follows from the pasting law of pushouts applied to thefollowing diagram gl (cid:101) R gl R gl R gl R ⊕ (1 + M [1]) 1 + M [1] . (cid:3) Hence, we are reduced to prove our result in the case of trivial square-zeroextensions. Next step will be to reduce to square-zero extensions of S . ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 17 Lemma 1.33
Let R be an E ∞ -ring, and let R ⊕ M be a trivial square-zero extension by an R -module M . Then, if Theorem 1.30 holds for trivialsquare-zero extensions of S , it holds for R ⊕ M → R . Proof.
Let R be an E ∞ -ring, and let S → R be the unit morphism. Givenany R -module M , this morphism induces by Remark 1.21 a Cartesian square S ⊕ M R ⊕ M S R (3)in the ∞ -category CAlg. The functor gl , being right adjoint, sends thesquare (3) to a co/Cartesian square in Sp: gl ( S ⊕ M ) gl ( R ⊕ M ) gl S gl R. (4)Let us denote by P the fiber of gl ( R ⊕ M ) → gl R . Then, P fits into thediagram: P gl ( S ⊕ M ) gl ( R ⊕ M )0 gl S gl R where the map P → gl ( S ⊕ M ) is induced by the fact that (4) is Cartesian.As the outer square is Cartesian by construction, it follows from the pastinglaw that P is canonically equivalent to the fiber of gl ( S ⊕ M ) → gl S , whichby hypothesis is equivalent to M . (cid:3) Lemma 1.34
Given any spectrum M , let 1 + M denote the fiber of themap gl ( S ⊕ M ) → gl S obtained by applying gl to the square-zero extension S ⊕ M → S . Then,the functor M (cid:55)→ M admits a left adjoint. Proof.
For any spectrum X , giving a map X → M is equivalent to give amap X → gl ( S ⊕ M ) together with a nullhomotopy for its postcompositionwith the map gl ( S ⊕ M ) → gl S . Upon passing to adjoints we see that, byvirtue of the explicit description of the augmentation map of S [ − ] given in ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 18 Remark 1.11, it is equivalent to give a map S [Ω ∞ a X ] S ⊕ M S that is, a morphism in Map CAlg / S ( S [Ω ∞ a X ] , S ⊕ M ). We can rephrase whatwe just observed more precisely, by saying that the functor M (cid:55)→ M fitsin the following diagram CAlg / S SpSp L S (cid:97) (cid:97) B ∞ (cid:71) S ⊕− (cid:97) S [Ω ∞ a − ] (5)i.e. if we denote by (cid:71) the right adjoint to the factorization of S [Ω ∞ a − ]through CAlg / S given by Remark 1.9, then M (cid:55)→ M is given by thecomposite right adjoint B ∞ (cid:71) ( S ⊕ − ). Hence, the functor M (cid:55)→ M isright adjoint to the functor X (cid:55)→ L S ( S [Ω ∞ a X ]). (cid:3) Remark 1.35
We stress that in (5) both S [Ω ∞ a − ] and B ∞ (cid:71) are functorsfrom the slice ∞ -category CAlg / S ; in particular, it is important to make adistinction between B ∞ (cid:71) and gl .1.4. Groups of units of square-zero extensions, top-down.
We arenow ready to give a first proof.
Proof of Theorem 1.30.
By virtue of Proposition 1.32 and Lemma 1.33, allit is left to do, it is to prove the theorem for trivial square-zero extensions S ⊕ M → S where M is a connective spectrum. As we want to show that 1 + M isnaturally equivalent to M , it is enough to prove that the functor X (cid:55)→L S ( S [Ω ∞ a X ]), left adjoint to M (cid:55)→ M , is naturally equivalent to theidentity functor. But, by virtue of Proposition 1.25, we know that this isindeed the case. (cid:3) Groups of units of square-zero extensions, bottom-up.
It ispossible to give an alternative proof of (the last step of the proof of) Theorem1.30, based on space level arguments and Proposition A.3. We dedicatethis section to give such an alternative proof. The idea is to show theequivalence of the spaces Ω ∞ M and Ω ∞ (1 + M ), and then to show thereare no obstructions to lift the comparison map at the level of connectivespectra. We begin with the following observation. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 19 Proposition 1.36
Let R be a connective E ∞ -ring, and let (cid:101) R → R be asquare-zero extension of R by a connective R -module M . Let us denoteby 1 + M the fiber of the induced map gl (cid:101) R → gl R . Then, the spacesΩ ∞ (1 + M ) and Ω ∞ M are equivalent in the ∞ -category S of spaces. Proof.
By applying Ω ∞ to the co/fiber sequence1 + M → gl (cid:101) R → gl R we get a fiber sequence of pointed spacesΩ ∞ (1 + M ) → GL (cid:101) R → GL R. By the commutativity of the following diagram in the ∞ -category S ∗ Ω ∞ (1 + M ) GL (cid:101) R Ω ∞ M Ω ∞ (cid:101) R pt GL R pt Ω ∞ R Sh ( ι (cid:101) R ) Sh ( ι R ) (where Sh denotes the basepoint shift of Definition 1.14, and ι R denotes themap given in Remark 1.7) we get a pointed map ψ : Ω ∞ (1+ M ) → Ω ∞ M . Itfollows from Proposition 1.16 that we have the following induced morphismbetween the homotopy fiber exact sequences · · · π k GL (cid:101) R π k GL R π k − (1 + M ) π k − GL (cid:101) R π k − GL R · · ·· · · π k (cid:101) R π k R π k − M π k − (cid:101) R π k − R · · · (cid:39) (cid:39) (cid:39) (cid:39) (6)which, by the five lemma, induces isomorphisms π k (1 + M ) (cid:39) π k M for k ≥
1. To conclude, wee need to prove that π ( ψ ) : π (1 + M ) → π M is anisomorphism. It follows from the exactness of the sequence · · · π (cid:101) R π R π M π (cid:101) R π R ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 20 that the map π R → π M is actually the zero map. If we consider (6) inthe case k = 1, we get · · · π (cid:101) R π R π (1 + M ) ( π (cid:101) R ) × ( π R ) × · · · π k (cid:101) R π R π M π (cid:101) R π R . (cid:39) (cid:39) The five-lemma implies that π (1 + M ) → π M is a monomorphism, whichin turn tells us that the map π R → π (1 + M ) is also the zero map. Hence, π (1 + M ) is the kernel of ( π (cid:101) R ) × → ( π R ) × and by Proposition A.3, π (1 + M ) (cid:39) π M , where one isomorphism is givenby u (cid:55)→ u −
1. Finally, it follows from the definition of Sh that the map π ( ψ ) : π (1 + M ) → π M is exactly given by u (cid:55)→ u − (cid:3) We can specialize the previous proposition to the case of trivial square-zero extensions of S , obtaining the following corollary. Corollary 1.37
The functor Ω ∞ ◦ (1 + − ) : Sp cn → S is equivalent to thefunctor Ω ∞ .We conclude this section with the promised second proof. (Alternative) Proof of Theorem 1.30. Again, keeping in mind Proposition1.32 and Lemma 1.33, we just have to show that the functor 1 + − describedin Lemma 1.34 is equivalent to the identity functor. It follows from [GGN15,2.10] and the equivalence Sp cn (cid:39) CGrp that there is an equivalenceFun Π (Sp cn , Sp cn ) (cid:39) Fun Π (Sp cn , S )(where Fun Π ( (cid:67) , (cid:68) ) denotes the full subcategory of Fun( (cid:67) , (cid:68) ) spanned byproduct preserving functors) given by postcomposition with Ω ∞ . By Lemma1.34, the functor 1 + − preserves all limits (in particular, products), andby Corollary 1.37, its postcomposition with Ω ∞ is equivalent to Ω ∞ itself.As this is also true for the identity functor, 1 + − and id Sp cn must beequivalent. (cid:3) Logarithmic derivatives of E ∞ -rings Let R be an ordinary commutative ring, and let M be an R -module.Given a derivation ∂ : R → M , the functionlog ∂ : R × → Mr (cid:55)→ ∂ ( r ) r − is easily seen to be a group homomorphism (from the group of units of R tothe underlying additive Abelian group of M ), and it is called the logarithmic ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 21 derivative relative to ∂ . Our goal for this section, is to construct a homotopycoherent analogue of logarithmic derivatives for E ∞ -rings.To this end, let now R be a connective E ∞ -ring (see Remark 1.31). andlet ∂ : L R → M be a derivation from R to an R -module M . Let us denoteby (cid:101) ∂ : R → M the composite map R d ∂ −→ R ⊕ M pr −−→ M in the ∞ -category Sp (where d ∂ is the map adjoint to ∂ , and pr is theprojection on the module part of the square-zero extension R ⊕ M ). As afirst approach, it is possible to mimic in a straightforward way the ordinarydefinition at the level of spaces, and thus to consider the compositionGL R ∆ −→ GL R × GL R ( ι R ◦ inv) × (Ω ∞ (cid:101) ∂ ◦ ι R ) −−−−−−−−−−−−→ Ω ∞ R × Ω ∞ M a −→ Ω ∞ M (7)in the ∞ -category of spaces (where ι R is as in Remark 1.7, inv is the inver-sion map given by the E ∞ -group structure of GL R , and a is given by theaction of R on M ). Such a map can be shown to be a morphism of groupobjects in the homotopy category ho S (i.e. a map of H-spaces). Our goalwill be to promote such a map to a map of E ∞ -groups, or to be more pre-cise, to produce a morphism of connective spectra whose underlying map ishomotopic to (7). In order to do so, we will exploit Theorem 1.30, appliedto the trivial square-zero extension R ⊕ M . Construction 2.1
Let R be a connective E ∞ -ring, and let ∂ : L R → M bea derivation of R into an R -module M . Theorem 1.30 implies that thereexists a co/fiber sequence M → gl ( R ⊕ M ) → gl R in the ∞ -category Sp, which splits, since R ⊕ M is a trivial square-zeroextension; that is, we have that gl ( R ⊕ M ) (cid:39) ( gl R ) ⊕ M (8)in the ∞ -category Sp. Let us momentarily denote by i : gl R → gl ( R ⊕ M )the direct summand inclusion. Let now d ∂ : R → R ⊕ M be the map adjoint to ∂ (which, by definition, is a section of the trivialsquare-zero extension R ⊕ M → R ). We will denote by + log ∂ : gl R → ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 22 gl ( R ⊕ M ), the composition gl R gl R ⊕ gl R gl ( R ⊕ M ) ⊕ gl ( R ⊕ M ) gl ( R ⊕ M ) ∆ + log ∂ ( i ◦ inv) ⊕ gl ( d ∂ ) µ (where µ is the multiplication map induced by Remark B.22). Remark 2.2
Let us denote by p : gl ( R ⊕ M ) → gl R the projection tothe direct summand given by (8). The composition gl R + log ∂ −−−−→ gl ( R ⊕ M ) p −→ gl R is nullhomotopic. In fact, by construction, the triangle gl R gl ( R ⊕ M ) gl R i ◦ invinv p commutes. On the other hand, by definition of d ∂ , the map gl ( d ∂ ) fits intothe commutative triangle gl R gl ( R ⊕ M ) gl R. gl ( d ∂ ) p Since, as it follows e.g. from Remark B.22, µ ◦ ( p ⊕ p ) (cid:39) p ◦ µ , we have that p ◦ + log ∂ = p ◦ µ ◦ ( i ◦ inv ⊕ gl ( d ∂ )) ◦ ∆ (cid:39) µ ◦ ( p ⊕ p ) ◦ ( i ◦ inv ⊕ gl ( d ∂ )) ◦ ∆ (cid:39) µ ◦ ( p ◦ i ◦ inv) ⊕ ( p ◦ gl ( d ∂ )) ◦ ∆ (cid:39) µ ◦ (inv ⊕ id) ◦ ∆which, again by Remark B.22, is nullhomotopic. Definition 2.3
Let R be a connective E ∞ -ring, and ∂ : L R → M a deriva-tion. We define the logarithmic derivative log ∂ induced by ∂ as the map ofspectra log ∂ : gl R → M ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 23 induced by the map + log ∂ given by Construction 2.1, and the universalproperty of fibers, applied by virtue of Remark 2.2; i.e. log ∂ is the essentiallyunique map rendering the following diagram commutative gl RM gl ( R ⊕ M ) gl R. log ∂ + log ∂ Remark 2.4
By virtue of Remark B.21, upon applying Ω ∞ a to the loga-rithmic derivative, we obtain a map of E ∞ -groupsΩ ∞ a log ∂ : GL R → Ω ∞ a M whose underlying map of spaces is homotopic to the map (7) given in theintroduction to this chapter.To see this, let us begin by applying the natural transformation ι : GL ⇒ Ω ∞ m given in Remark 1.7 to d ∂ , in order to get a commutative squareGL R Ω ∞ m R GL ( R ⊕ M ) Ω ∞ m ( R ⊕ M ) ι R GL ( d ∂ ) Ω ∞ m ( d ∂ ) ι R ⊕ R in the ∞ -category of E ∞ -groups. If we consider the above square in the ∞ -category S , combined with (8), we get the followingGL R Ω ∞ R GL R × Ω ∞ M Ω ∞ R × Ω ∞ M Ω ∞ M Ω ∞ M ι R GL ( d ∂ ) Ω ∞ ( d ∂ ) ι R ⊕ M pr pr (where pr is the obvious projection) showing thatΩ ∞ (cid:101) ∂ ◦ ι R (cid:39) pr ◦ Ω ∞ ( d ∂ ) ◦ ι r (cid:39) pr ◦ GL ( d ∂ ) (9)as map of spaces. Now, by (8), Ω ∞ a log ∂ is homotopic toGL R Ω ∞ a ( + log ∂ ) −−−−−−−→ GL ( R ⊕ M ) pr −−→ Ω ∞ a M. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 24 Unraveling the definitions, we have that, in the ∞ -category S :Ω ∞ log ∂ (cid:39) pr ◦ Ω ∞ ( + log ∂ ) (cid:39) pr ◦ Ω ∞ (cid:16) µ ◦ ( i ◦ inv) ⊕ gl ( d ∂ ) ◦ ∆ (cid:17) (cid:39) pr ◦ (cid:101) m ◦ Ω ∞ (cid:16) i ◦ inv ⊕ gl ( d ∂ ) (cid:17) ◦ ∆ (cid:39) pr ◦ (cid:101) m ◦ (cid:16) Ω ∞ ( i ◦ inv) × GL ( d ∂ ) (cid:17) ◦ ∆(where (cid:101) m denotes the multiplication map on GL ( R ⊕ M )). By Remark1.18, this is homotopic to m ◦ (cid:16) ( ι R ◦ inv) × (pr ◦ GL ( d ∂ )) (cid:17) ◦ ∆which in turn, by (9), is homotopic to (7).We conclude this chapter showing that for ordinary rings, regarded asdiscrete E ∞ -rings, our definition of logarithmic derivatives recovers the usualone, upon passing to connected components. Remark 2.5 If R is a discrete E ∞ -ring, [Lur17, 7.4.3.8] shows that π L R (cid:39) Ω π R in the ordinary category of discrete π R -modules. As a consequence, anyderivation ∂ : L R → M determines an ordinary derivation π R → π M . Proposition 2.6
Let R be a discrete E ∞ -ring, and let ∂ : L R → M be aderivation of R into an R -module M . Then, the morphism π log ∂ : ( π R ) × → π M is the ordinary logarithmic derivative associated to π ∂ . Proof.
Let us denote by (cid:101) ∂ : π R → π M the ordinary derivation determinedby π ∂ : Ω π R → π M . Unraveling the definitions, we see that the value of π log ∂ : ( π R ) × → π ( gl ( R ⊕ M )) (cid:39) ( π R ) × ⊕ π M on any element r ∈ ( π R ) × is π log ∂ ( r ) = π (cid:16) µ ◦ (cid:0) i ◦ inv ⊕ gl ( d ∂ ) (cid:1) ◦ ∆ (cid:17) ( r )= π (cid:16) µ ◦ ( i ◦ inv ⊕ gl ( d ∂ )) (cid:17) ( r, r )= π ( µ ) (cid:16)(cid:0) r − , (cid:1) , (cid:0) r, (cid:101) ∂ ( r ) (cid:1)(cid:17) = (cid:16) r − , (cid:17) · (cid:16) r, (cid:101) ∂ ( r ) (cid:17) = (cid:16) , (cid:101) ∂ ( r ) r − (cid:17) ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 25 where the last equality follows from Remark 1.18. Now, it follows fromProposition A.3 that the induced map π log ∂ : ( π R ) × → π M is just theprojection on the second factor of π log ∂ , and hence it is given on elementsby r (cid:55)→ (cid:101) ∂ ( r ) r − . (cid:3) Appendix A. The classical result
Our goal in this appendix is to recall and prove Proposition A.3, whichis the ordinary version of Theorem 1.30. Throughout by “ring” we meancommutative ring with unit; by “ring (homo)morphism” we mean unit-preserving ring homomorphism.
Definition A.1
Given a surjective morphism of rings ϕ : (cid:101) R → R the morphism ϕ is said to be a square-zero extension if (ker ϕ ) = 0.With a little abuse of terminology, we will say that “ (cid:101) R is a square-zeroextension of R by I ” if R and (cid:101) R are rings, and R (cid:39) (cid:101) R/I for some ideal I ⊂ R such that I = 0. Remark A.2
Let ϕ : (cid:101) R → R be a square-zero extension and let I := ker ϕ denote its kernel. Then we have a short exact sequence of (cid:101) R -modules0 −→ I −→ (cid:101) R −→ R −→ . Proposition A.3
Given a square-zero extension ϕ : (cid:101) R → R , let I := ker ϕ denote its kernel. Then there exists an induced short exact sequence ofgroups 0 −→ I ι −→ GL (cid:101) R (cid:101) ϕ −→ GL R −→ ι ( r ) = 1 + r ; (cid:101) ϕ ( r ) = ϕ ( r ) . Proof.
First we need to check that everything is well defined. Since byhypothesis we have that ϕ is a square-zero extension, then ι ( r ) ι ( − r ) = (1 + r )(1 − r ) = 1hence ι ( − r ) = ι ( r ) − ; again, by hypothesis ι ( r ) ι ( r (cid:48) ) = (1 + r )(1 + r (cid:48) ) = 1 + r + r (cid:48) = ι ( r + r (cid:48) ) , thus ι is a group homomorphism. Since ϕ is a ring homomorphism, itpreserves units, therefore (cid:101) ϕ is well defined. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 26 The sequence is also exact. Clearly, ι is injective. To see that ϕ reflectsunits, let a, b ∈ GL R be such that ab = 1, and let α, β ∈ (cid:101) R be such that ϕ ( α ) = a and ϕ ( β ) = b . It follows from ϕ ( αβ −
1) = ab − αβ − ∈ I ; hence, as ( αβ − = 0 we have αβ (2 − αβ ) = 1proving that both α and β are units in (cid:101) R . Now, sinceker (cid:101) ϕ = (cid:101) ϕ − ( { } ) = ϕ − ( { } ) = 1 + I = ι ( I )the sequence is exact. (cid:3) Appendix B. Fundamentals of higher commutative algebra
B.1.
Symmetric monoidal ∞ -categories.Notation B.1 We denote by E ⊗∞ the ∞ -category N(Fin ∗ ), that is, the nerveof the category of finite pointed sets. Given any n ∈ N , and any 1 ≤ i ≤ n ,we denote by ρ i : (cid:104) n (cid:105) → (cid:104) (cid:105) the function sending all elements of (cid:104) n (cid:105) to (thebasepoint) 0, with the exception of the element i . Definition B.2 A symmetric monoidal ∞ -category is the datum of an ∞ -category (cid:67) ⊗ together with a coCartesian fibration of simplicial sets p : (cid:67) ⊗ → E ⊗∞ satisfying the following “Segal condition”. • For every n ≥
0, and every 0 ≤ i ≤ n the functors ρ i ! : (cid:67) ⊗(cid:104) n (cid:105) → (cid:67) ⊗(cid:104) (cid:105) induced by the functions ρ i and the coCartesian fibration p , deter-mine an equivalence (cid:0) ρ i ! (cid:1) ni =1 : (cid:67) ⊗(cid:104) n (cid:105) ∼ −→ n (cid:89) i =1 (cid:67) ⊗(cid:104) (cid:105) . We denote by (cid:67) the ∞ -category (cid:67) ⊗(cid:104) (cid:105) , and, slightly abusing terminology, wesay that p exhibits a symmetric monoidal structure on (cid:67) , and that (cid:67) itselfis a symmetric monoidal ∞ -category .We refer the reader to [Lur17, Chapter 2] for a discussion about howthis definition gives a homotopy coherent generalization of the ordinarynotion of a symmetric monoidal category. In particular, if (cid:67) ⊗ → E ⊗∞ isa symmetric monoidal structure on (cid:67) , it follows from the definitions thatthere exists a uniquely (up to canonical isomorphism) determined bifunctor,denoted ⊗ : (cid:67) × (cid:67) → (cid:67) , encoded by the symmetric monoidal structure(see [Lur17, 2.0.0.6, 2.1.2.20]). ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 27 Definition B.3
Let (cid:67) be a symmetric monoidal ∞ -category. We say that (cid:67) is closed if, for each C ∈ (cid:67) , the functor (cid:67) (cid:39) (cid:67) × ∆ × C −−−→ (cid:67) × (cid:67) ⊗ −→ (cid:67) (informally given by D (cid:55)→ D ⊗ C ) admits a right adjoint. Definition B.4
Let (cid:67) and (cid:68) be symmetric monoidal ∞ -categories withsymmetric monoidal structures p : (cid:67) ⊗ → E ⊗∞ and q : (cid:68) ⊗ → E ⊗∞ .(1) A lax symmetric monoidal functor is given by a map of ∞ -operads F : (cid:67) ⊗ → (cid:68) ⊗ (i.e. a morphism of ∞ -categories over E ⊗∞ , carry-ing p -coCartesian lifts of inert morphisms of E ⊗∞ to q -coCartesianmorphisms in (cid:68) ⊗ ).(2) A symmetric monoidal functor is given by a morphism of ∞ -cate-gories over E ⊗∞ carrying p -coCartesian morphisms to q -coCartesianmorphisms.We sometimes abuse notation, and refer to (lax) symmetric monoidalfunctors indicating only the underlying functor between the underlying ∞ -categories. Example B.5
We have the following two notable examples (see also Defi-nition B.19):(1) [Lur17, Section 2.4.1] Given any ∞ -category (cid:67) with finite products,it has a symmetric monoidal structure, denoted (cid:67) × → E ⊗∞ , encoding its Cartesian product.(2) [Lur17, 4.8.2.14] The ∞ -category S ∗ of pointed spaces has a sym-metric monoidal structure, denoted S ∧∗ → E ⊗∞ , encoding the smash product of pointed spaces. Definition B.6
Let (cid:67) be a symmetric monoidal ∞ -category. We letCAlg( (cid:67) ) denote the full subcategory of sections E ⊗∞ → (cid:67) ⊗ of the struc-ture map (cid:67) ⊗ → E ⊗∞ spanned by lax symmetric monoidal functors. Werefer to CAlg( (cid:67) ) as the ∞ -category of commutative algebra objects of (cid:67) . Definition B.7
Let (cid:67) be an ∞ -category with finite products. A commu-tative monoid object of (cid:67) is given by a functor M : E ⊗∞ → (cid:67) such that themorphisms M ( (cid:104) n (cid:105) ) → M ( (cid:104) (cid:105) ) induced by the inert morphisms (cid:104) n (cid:105) → (cid:104) (cid:105) exhibit M ( (cid:104) n (cid:105) ) as an n -fold product of M ( (cid:104) (cid:105) ). We let CMon( (cid:67) ) denotethe full subcategory of Fun( E ⊗∞ , (cid:67) ) spanned by the commutative monoidsof (cid:67) . ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 28 As it is clear, in an ∞ -category with finite products, it is possible to defineboth commutative algebra objects with respect to the Cartesian symmet-ric monoidal structure and commutative monoid objects; in fact, the twodefinitions agree. Proposition B.8 [Lur17, 2.4.2.5] Let (cid:67) be an ∞ -category with finite prod-ucts, considered as a symmetric monoidal ∞ -category with the Cartesianstructure of Example B.5.1. Then, there is an equivalence of ∞ -categoriesCAlg( (cid:67) ) (cid:39) CMon( (cid:67) ) . Remark B.9
A commutative algebra object A : E ⊗∞ → (cid:67) ⊗ of a symmetricmonoidal ∞ -category (cid:67) determines an object A ( (cid:104) (cid:105) ) of (cid:67) together witha homotopy coherent analogue of the structure of a commutative algebraobject of an ordinary symmetric monoidal category. In fact, the assignment A (cid:55)→ A ( (cid:104) (cid:105) ) extends to a forgetful functorCAlg( (cid:67) ) → (cid:67) . With a little abuse of notation, we often denote A ( (cid:104) (cid:105) ) just by A , and referto it as a commutative algebra object (or a commutative monoid object, ifthe symmetric monoidal structure on (cid:67) is Cartesian). Remark B.10
It follows from the definitions, that, if M is a commutativemonoid in an ∞ -category (cid:67) , then the underlying object M in the homotopycategory ho (cid:67) is a commutative monoid object (in the ordinary sense). Proposition B.11 [GGN15, 1.1] Let (cid:67) be an ∞ -category with finiteproducts, and let M be a commutative monoid object of (cid:67) . Then thefollowing conditions are equivalent:(1) M admits an inversion map for the multiplication induced by thecommutative algebra object structure.(2) The commutative monoid object of ho (cid:67) underlying M is a groupobject. Definition B.12
Let (cid:67) be an ∞ -category with finite products; we saythat a commutative monoid object M ∈ CMon( (cid:67) ) is a commutative groupobject if it satisfies the equivalent conditions of Proposition B.11. We writeCGrp( (cid:67) ) to denote the full subcategory of CMon( (cid:67) ) consisting of commu-tative group objects.We refer the reader to [GGN15] for other equivalent characterizations ofcommutative group objects in an ∞ -category, and for a detailed treatmentof some of its properties. Remark B.13
As we observed in the introduction, the ∞ -category ofspaces plays in the homotopy coherent world the same role that the ordinarycategory of sets plays in the ordinary case. Accordingly, we denote the ∞ -categories CMon( S ) and CGrp( S ) just by CMon and CGrp, respectively. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 29 We sometimes refer to CMon as the ∞ -category of E ∞ -spaces or as the ∞ -category of E ∞ -monoids , and we refer to CGrp as the ∞ -category of E ∞ -groups .Similarly, as the ∞ -category of spectra is the homotopy coherent analogueof the category of Abelian groups, we denote the ∞ -category CAlg(Sp) justby CAlg, and refer to it as the ∞ -category of E ∞ -rings .The difference between the ∞ -category of E ∞ -groups and the ∞ -categoryof spectra is something that has no counterpart in the ordinary case. As re-called below (see Remark B.21), the ∞ -category of E ∞ -groups is equivalentto the full subcategory Sp cn ⊂ Sp of connective spectra, which is not stable,and the smallest stable presentable ∞ -category containing it is precisely Sp.In a sense, looking for an homotopy coherent analogue of the Abelian cat-egory of Abelian groups, one can start with the more “naive” ∞ -categoryof E ∞ -groups (the straightforward generalization of the category of Abeliangroups), and then the price to pay to have stability (the generalization ofthe notion of being Abelian we want to work with) is to add nonconnectivespectra to the picture, which (as far as the analogy with the ordinary casegoes) can be thought of as a sort of technical nuisance. Proposition B.14
Let p : (cid:67) ⊗ → E ⊗∞ and q : (cid:68) ⊗ → E ⊗∞ be symmetricmonoidal ∞ -categories, and let F : (cid:67) ⊗ → (cid:68) ⊗ be a symmetric monoidalfunctor, such that the underlying functor F (cid:104) (cid:105) : (cid:67) → (cid:68) admits a right ad-joint. Then F admits a right adjoint G : (cid:68) ⊗ → (cid:67) ⊗ , which is lax symmetricmonoidal. Proof.
This follows immediately from [Lur17, 7.3.2.1]. (cid:3)
Adjunctions of the kind of Proposition B.14 constitute a homotopy co-herent analogue of ordinary symmetric monoidal adjunctions, hence we willadopt the same terminology in this context.
Definition B.15
Let p : (cid:67) ⊗ → E ⊗∞ and q : (cid:68) ⊗ → E ⊗∞ be symmetricmonoidal ∞ -categories. We say that an adjunction F : (cid:67) ⊗ (cid:29) (cid:68) ⊗ : G is a symmetric monoidal adjunction if the left adjoint is symmetric monoidaland the right adjoint is lax symmetric monoidal.We now pose our attention to the case of presentable ∞ -categories. Proposition B.16 [GGN15, 4.1, 4.4] Given a presentable ∞ -category (cid:67) ,the ∞ -categories CMon( (cid:67) ) and CGrp( (cid:67) ) are presentable; moreover, thereare functors (cid:67) → CMon( (cid:67) ) → CGrp( (cid:67) )which are left adjoint to the respective forgetful functors. ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 30 Recall that, given a presentable ∞ -category (cid:67) , we can define its stabi-lization Exc ∗ ( S fin ∗ , (cid:67) ) (which is again presentable), denoted Sp( (cid:67) ), relatedto (cid:67) by an adjunction Σ ∞ + : (cid:67) (cid:29) Sp( (cid:67) ) : Ω ∞ where the left adjoint is called the suspension spectrum functor (see [Lur17,Section 1.4.2] for details). Proposition B.17 [GGN15, 4.10] Let (cid:67) be a presentable ∞ -category.The suspension spectrum functor Σ ∞ + : (cid:67) → Sp( (cid:67) ) factors as a compositionof left adjoints (cid:67) ( − ) + −−−→ (cid:67) ∗ F −→ CMon( (cid:67) ) ( − ) gp −−−→ CGrp( (cid:67) ) B ∞ −−→ Sp( (cid:67) )each of which is uniquely determined by the fact that it commutes with thecorresponding free functor from (cid:67) .When (cid:67) is a presentable symmetric monoidal ∞ -category, the abovechain of adjunctions can be enhanced to a chain of symmetric monoidaladjunctions. Proposition B.18 [GGN15, 5.1] Let (cid:67) be a presentable closed symmet-ric monoidal ∞ -category. The ∞ -categories (cid:67) ∗ , CMon( (cid:67) ), CGrp( (cid:67) ) andSp( (cid:67) ) all admit closed symmetric monoidal structures, which are uniquelydetermined by the requirement that the respective free functors from (cid:67) aresymmetric monoidal. Moreover, each of the functors (cid:67) ∗ → CMon( (cid:67) ) → CGrp( (cid:67) ) → Sp( (cid:67) )uniquely extends to a symmetric monoidal left adjoint.
Definition B.19
The stabilization Sp( S ) of the ∞ -category of spaces is the ∞ -category of spectra Sp. The product encoded by the symmetric monoidalstructure Sp ⊗ → E ⊗∞ induced on Sp by Proposition B.18 is commonly referred to as the smashproduct of spectra . Notation B.20
In most cases, we omit notations for the forgetful functors.In the special case (cid:67) = S , we adopt the following notations and terminology:(1) We denote the right adjoint to B ∞ as Ω ∞ a and refer to it as the underlying additive E ∞ -space functor.(2) We denote the composite left adjoint B ∞ ◦ ( − ) gp ◦ F byΣ ∞ : S ∗ → Spand denote its right adjoint, with the usual abuse of notation, byΩ ∞ . ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 31 (3) We denote the composite left adjoint F ◦ ( − ) + by F + : S →
CMon . Remark B.21 [Lur17, 5.2.6.26] The ∞ -category CGrp of E ∞ -groups andthe ∞ -category Sp cn of connective spectra are equivalent. The equivalenceis given by the functor B ∞ defined above, whose essential image is exactlySp cn . In particular, the composite functorSp Ω ∞ a −−→ CGrp ∼ −→ Sp cn is equivalent to the truncation functor τ ≥ , right adjoint to the inclusionSp cn → Sp.
Remark B.22
As Sp is a stable ∞ -category, [Lur17, 1.1.3.5] implies thatit is also an additive ∞ -category (see [GGN15, Section 2] for the definitionand a detailed discussion of this property). Therefore, by [GGN15, 2.8],every spectrum admits a commutative group structure, and all maps be-tween spectra respect this structure; to be more precise, the forgetful mapCGrp(Sp) → Sp is an equivalence. In particular, given any spectrum X ,there exists a map inv : X → X such that the composition X ∆ −→ X ⊕ X id ⊕ inv −−−−→ X ⊕ X µ −→ X (where ∆ is the diagonal map, and µ is the multiplication induced by thecommutative algebra structure on X ) is nullhomotopic.The above remark is consistent with the fact that we think of Sp as thehomotopy coherent analogue of the Abelian category of Abelian groups, andcan be interpreted as saying that all spectra admit a “homotopy coherentcommutative group structure”.B.2. Modules over commutative algebra objects.
Let (cid:67) be a symmet-ric monoidal ∞ -category, and let R ∈ CAlg( (cid:67) ) be a commutative algebraobject in (cid:67) . Analogously to what happens in the ordinary case, it is pos-sible to define an ∞ -category Mod R ( (cid:67) ) of R -modules which, under mildassumptions, is again symmetric monoidal. In [Lur17, 3.3, 4.5] a few equiv-alent models for this ∞ -category are constructed, together with a detaileddiscussion of their equivalence. For future reference, we recollect in thissection some results about these ∞ -categories, whose formulation (and va-lidity) are independent of the specific model, referring the reader to [Lur17]for details and proofs. Notation B.23
Let (cid:67) be a symmetric monoidal ∞ -category, and let R ∈ CAlg( (cid:67) ) be a commutative algebra object in (cid:67) . We denote by Mod R ( (cid:67) )the ∞ -category of R -modules . If (cid:67) = Sp, we denote Mod R ( (cid:67) ) just byMod R . ∞ -GROUPS OF UNITS AND LOGARITHMIC DERIVATIVES 32 Proposition B.24 [Lur17, 4.5.2.1] Let (cid:67) be a symmetric monoidal ∞ -cat-egory. Assume that (cid:67) admits geometric realizations of simplicial objects,and that the tensor product ⊗ : (cid:67) × (cid:67) → (cid:67) preserves geometric realizationsof simplicial objects separately in each variable. Let R ∈ CAlg( (cid:67) ) be acommutative algebra object in (cid:67) . The ∞ -category Mod R ( (cid:67) ) admits asymmetric monoidal structureMod R ( (cid:67) ) ⊗ R → E ⊗∞ whose symmetric monoidal product is also called the relative tensor product . Proposition B.25
Let (cid:67) be a symmetric monoidal ∞ -category as inProposition B.24. Let A and B be commutative algebra objects in (cid:67) , andlet f : A → B be a morphism in CAlg( (cid:67) ). Then, there exists a symmetricmonoidal adjunction − ⊗ A B : Mod A ( (cid:67) ) (cid:29) Mod B ( (cid:67) ) : f ! with the left adjoint given by the relative tensor product M (cid:55)→ M ⊗ A B .We refer to such left adjoint as the extension of scalars functor, and to theright adjoint as the restriction of scalars functor. Proof.
The existence of the adjunction on the underlying ∞ -categories fol-lows from [Lur17, 4.6.2.17], which also implies that the left adjoint is givenby the relative tensor product. By [Lur17, 4.5.3.2], the left adjoint is sym-metric monoidal. Finally, it follows from [Lur17, 7.3.2.7] that the rightadjoint is lax monoidal. (cid:3) We sometimes omit notation for the restriction of scalars functor.
Remark B.26
It follows from [Lur17, 4.8.2.19] that the smash product ⊗ : Sp × Sp → Sppreserves small colimits separately in each variable; in particular, Sp satisfiesthe hypotheses of Proposition B.24.In ordinary commutative algebra, the category of modules over a ring isAbelian. Analogously, the ∞ -category of modules over an E ∞ -ring is stable. Proposition B.27 [Lur17, 7.1.1.5] Let R be an E ∞ -ring. Then, the ∞ -category Mod R of R -modules is stable. References [ABG +
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