aa r X i v : . [ m a t h . A T ] F e b Adjoining roots in homotopy theory
Tyler LawsonFebruary 7, 2020
Abstract
We use a “twisted group algebra” method to constructively adjoin formalradicals n √ α , for α a unit in a commutative ring spectrum or an invertibleobject in a symmetric monoidal ∞ -category. We show that this construction isclassified by maps from Eilenberg–Mac Lane objects to the unit spectrum gl ,the Picard spectrum pic , and the Brauer spectrum br . Given a commutative ring R and an element α ∈ R , we can adjoin a formal radical n √ α to R by embedding R into the extension ring R [ x ] / ( x n − α ) . These ring extensionscome equipped with a ready-made basis { , x, . . . , x n − } over R and are fundamentalconstructions in algebra. In the derived setting, however, it is less clear when thesetypes of constructions are possible. Given a commutative ring spectrum R and anelement in α ∈ π R , one can construct a commutative R -algebra with an n ’th root of α using the same type of presentation in terms of generators and relations. However,away from characteristic zero the universal property enjoyed by this construction isnot as strong and its coefficient ring can be unpredictable.We can try instead to lift the algebra directly by constructing a commutative R -algebra S with a map R → S such that, on coefficient rings, we have the algebraicextension: π ∗ S (cid:27) ( π ∗ R )[ x ] / ( x n − α ) . Depending on α , such an extension may not bepossible or may not be unique. It is always possible to adjoin roots of , becausethose algebras are realized by the group algebras R [ C n ] of finite cyclic groups. If both α and n are units in π R then the resulting extension on coefficient rings is étale , andthe obstruction theory of Robinson [Rob03] or Goerss–Hopkins [GH04] can be usedto show that the extension ring S exists, is essentially unique, and has a universalproperty among R -algebras with such a root adjoined. If is invertible, this meansthat we can adjoint √− . However, Schwänzl–Vogt–Waldhausen showed in [SVW99],using topological Hochschild homology, that it is impossible to adjoint a square rootof − to the sphere spectrum. A different argument of Hopkins with K (1) -localpower operations shows that the p -complete K -theory spectrum cannot admit p ’throots of unity [LN14, A.6.iii], and this was further generalized by Devalapurkar to K ( n ) -local theory [Dev17]. Further difficulties appear when attempting to adjoin aroot that appears in a nonzero degree. 1ur starting observation is that these formal radicals are a special case of twistedgroup algebras. Given an abelian group extension → G → E → A → and agroup homomorphism G → R × , we can form the relative tensor product Z [ E ] ⊗ Z [ G ] R .The elements of A lift to a basis over R , and the resulting algebra differs from thegroup algebra R [ A ] by this central extension. Moveover, all such extensions can beconstructed in a universal case by pushing forward the extension from G to R × .This process can be applied to ring spectra. We will begin by showing that,when it is possible to construct similar extensions of the spectrum of units gl ( R ) ,we can give systematic constructions of formal radicals and other twisted groupalgebras, and obstructions are detectable with sufficient knowledge of gl ( R ) . Bythen reinterpreting these constructions as Thom spectra for maps to pic ( R ) , we canuse recent work of Antolín-Camarena–Barthel [AB14] to both generalize this andallow us to identify these algebras as having a universal property. This recoversseveral constructions: adjoining n ’th roots of elements to a ring spectrum where n is invertible, usually carried out using obstruction thery, and adjoining similar rootsto elements in gradings outside zero. There are also new constructions: we find thatwe can extend the first Postnikov stage τ ≤ S (2) of the -local sphere by adjoining √ D for D ≡ mod .Once we have accomplished this, our second goal will be to dig one categoricallevel down.The same methods can be applied to adjoin formal radicals of elements in thePicard group. This allows us to take an extension of the Picard spectrum pic ( R ) and use it to embed the category of R -modules into a graded category with alarger group of invertible objects. This formalism recovers algebraic examples, suchas Rezk’s ω -twisted tensor product for Z / -graded modules [Rez09]. There arealso new topological examples: if R is an MU -algebra we can embed the category LMod R of R -modules, where integer suspensions are possible, into a larger category Q Q /Z LMod R with a symmetric monoidal structure that allows suspensions by ele-ments of Q × Z / . This is also possible for modules over the topological K -theoryspectrum ku and the algebraic K -theory spectrum K C . Although our focus is onring spectra, many of the results are proved in the generality of presentable symmet-ric monoidal ∞ -categories.We finally close with the observation that algebraic K -theory relates these twoprocesses. Further directions
A first issue is that our discussion of adjoining roots is less satisfying for units outsidedegree zero. In particular, the identification of such units is somewhat roundabout.Ideally a solution to this problem would make use of a spectrum of graded unitssimilar to those developed by Sagave [Sag16], and in particular his construction of bgl ∗ R .Second, we restrict our attention to strictly commutative objects (meaning E ∞ -ring objects). The constructions in this paper should have interesting and useful E n -variants, making use of the iterated classifying spaces B ( n ) GL R .2inally, we study unit groups because they are more easily analyzed via theirassociated spectrum. This means that we lose any ability to extract formal radicalsof nonunit elements. We are hopeful that the future will bring a better understandingof the structure theory of E ∞ -spaces, allowing us to move beyond unit groups toeffectively study multiplicative monoids. Conventions and background
Our paper is written homotopically, and in particular we use the phrase “commuta-tive ring spectrum” to mean an E ∞ -ring spectrum.For expedience, we will use the same name for both an abelian group A and theassociated Eilenberg–Mac Lane spectrum, regarding the category of abelian groupsas embedded fully faithfully into the higher category of spectra. A commutative ring k gives rise to a commutative ring spectrum.For a ring spectrum R , the unit group GL ( R ) ⊂ Ω ∞ R is the space of units underthe multiplicative monoidal product, or equivalently the space of self-equivalences of R as a left R -module. If R has a commutative ring structure we write gl ( R ) for theassociated spectrum of units [May77, ABG + S [GL ( R )] → R for the adjunction between unit groups and spherical group algebras.For a monoidal ∞ -category C , the Picard space Pic( C ) ⊂ C ≃ is the space ofinvertible objects and equivalences between them [Cla11, MS16]. If C has a symmetricmonoidal structure then Pic( C ) has an E ∞ -structure and we write pic ( C ) for theassociated Picard spectrum. If C = LMod R is the category of modules over R , wesimply write Pic( R ) and pic ( R ) instead.We will require known identifications of the groups [ A, Σ k B ] of homotopy classesof maps between Eilenberg–Mac Lane spectra, which we will simply state [EML54].
1. The group [ A, B ] is isomorphic to the group Hom(
A, B ) .2. The group [ A, Σ B ] is isomorphic to the group Ext(
A, B ) : this extension isidentified with the fiber of a map A → Σ B .3. The group [ A, Σ B ] is isomorphic to the group Hom(
A, B [2]) of -torsionhomomorphisms A → B .4. The group [ A, Σ B ] is part of a short exact sequence → Ext(
A, B [2]) → [ A, Σ B ] → Hom(
A, B/ → .
5. These identifications respect composition. In particular, the composition map [ B, Σ C ] × [ A, Σ B ] → [ A, Σ C ] is the Yoneda pairing Hom(
B, C [2]) × Ext(
A, B ) → Ext(
A, C [2]) . (cf. [Mat14, §2.2]) If the unit of C is κ -compact for some cardinal κ , then the objects of Pic( C ) arealso κ -compact, and if C is presentable then the full subcategory of κ -compact objects is essentially small.Therefore, for presentable monoidal ∞ -categories it is possible to identify Pic( C ) with a small space eventhough C is large. All of these can be determined by first calculating that the groups of maps Z → Σ k Z are Z , , , Z / , for k = 0 ... , and then using free resolutions of the source and target. The generatorin degree is the composite of mod- reduction, the Steenrod square Sq , and the Bockstein.
3s a result, a pair ( g, Γ ) representing a homomorphism B → C [2] and anextension → B → Γ → A → maps to zero if and only if the homomorphism g extends from B to all of Γ .Given an ordinary symmetric monoidal category D , the Picard space Pic( D ) isthe nerve of an ordinary groupoid, consisting of the invertible objects and isomor-phisms between them. This means there is a fiber sequence pic ( D ) → π pic ( D ) k −→ Σ π pic ( D ) . Here π pic ( D ) is the classical Picard group of the category D , and π pic ( D ) is theautomorphism group Aut D ( I ) of the monoidal unit. This first k -invariant is alwaysexpressed in terms of the twist map. Namely, given an invertible module γ , thetwist-self-isomorphism γ ⊗ γ → γ ⊗ γ is multiplication by a 2-torsion automorphism τ ( γ ) of the monoidal unit, an element of π pic ( D ) that satisfies τ ( γ ) ◦ τ ( γ ) = id .The k -invariant π pic ( D ) k −→ Σ π pic ( D ) is identified with τ . Acknowledgements
The author would like to thank Tobias Barthel, Clark Barwick, Sanath Devalapurkar,Fabien Hebestreit, Lars Hesselholt, and Charles Rezk for discussions related to thispaper, and Benjamin Antieau for comments on an earlier draft. The author wouldalso like to thank the Max Planck Institute for their hospitality and financial supportwhile this paper was being written. The author was partially supported by NSF grant1610408.
Let α ∈ π ( R ) be a unit; α is then also represented by an element of π ( gl ( R )) . Fixa positive integer n . From this, we can construct an extension of abelian groups → π ( gl R ) → E → Z /n → such that the generator of Z /n has a chosen lift to an element x ∈ E with x n = α .As a set with an action of π gl ( R ) , E (cid:27) ` n − i =0 π gl ( R ) · { x i } .This extension is determined by an extension class Ext ( Z /n, π gl ( R )) , orequivalently by a map ¯ ρ : Z /n → Σ π gl ( R ) between Eilenberg–Mac Lane spectra. Definition 1. A formal n ’th root of α is a lift ρ of ¯ ρ to gl ( R ) : Σ gl ( R ) (cid:15) (cid:15) Z /n ρ ttttttttt ¯ ρ / / Σ π gl ( R )
4e refer to the fiber of ρ as the extended unit spectrum gl ( R, ρ ) associated to ρ , andthe associated infinite loop space as extended unit group GL ( R, ρ ) .The extended unit spectrum is part of a fiber sequence gl ( R ) → gl ( R, ρ ) → Z /n, and hence we get a decomposition GL ( R, ρ ) (cid:27) ` n − i =0 GL ( R ) · { x i } as spaces with anaction of GL ( R ) . Remark . The map ρ determines ¯ ρ : the map gl ( R ) → gl ( R, ρ ) is an isomorphismon π k except when k = 0 , when it is the inclusion π gl ( R ) → E . Therefore, ρ determines the extension E and hence determines α up to n ’th powers. Definition 3.
Suppose that ρ is a formal n ’th root of α . Then the algebra obtainedby adjoining this root is the relative smash product R [ ρ ] = S [GL ( R, ρ )] ⊗ S [GL ( R )] R. Proposition 4.
The coefficient ring of R [ ρ ] is π ∗ R [ ρ ] (cid:27) π ∗ R [ x ] / ( x n − α ) . Proof.
The decomposition GL ( R, ρ ) (cid:27) ` n − i =0 GL ( R ) · { x i } means that the spheri-cal group algebra S [GL ( R, ρ )] decomposes as ⊕ n − i =0 S [GL ( R )] · { x i } , a free left S [GL ( R )] -module. Therefore, there is a simple Künneth formula that gives usan isomorphism of modules: π ∗ R [ ρ ] (cid:27) π ∗ S [GL ( R, ρ )] ⊗ π ∗ S [GL ( R )] π ∗ R (cid:27) n − M i =0 π ∗ R · { x i } . Moreover, the identity x n = α for the element x ∈ E = π GL ( R, ρ ) completelydetermines the multiplication in π ∗ R [ ρ ] . Remark . It is clear that, other than the calculation of the structure of the coefficientring, there is nothing special about the group Z /n in the above discussion. Given amap ρ : A → Σ gl ( R ) for some abelian group A , lifting an extension → π gl ( R ) → E → A → , there is an associated algebra R [ ρ ] whose coefficient ring is a twisted central exten-sion: π ∗ R [ ρ ] (cid:27) Z [ E ] ⊗ Z [ π gl ( R )] π ∗ R (cid:27) M a ∈ A π ∗ R · { a } . We will see similar algebras in later sections.5 xample . Suppose that n is a unit in π R . Then n acts invertibly on the homotopygroups π k gl ( R ) (cid:27) π k R for k > : therefore the spectrum of maps from Z /n to the -connected cover τ ≥ gl ( R ) is trivial. Using the fiber sequence Σ gl ( R ) → Σ π gl ( R ) → Σ τ ≥ gl ( R ) , we find that, for any unit α , the map Z /n → Σ π gl ( R ) lifts essentially uniquely toa formal n ’th root Z /n → Σ gl ( R ) . Example . Let K be the p -complete K -theory spectrum. Then it is possible to showthat the group [ Z /p, Σ gl K ] is trivial, and thus this method does not allow us toadjoin x = p √ α for any nontrivial element α of Z × p / ( Z × p ) p . To give a proof, however,we need to know structural properties of the multiplication on K . The straight-lineproof we know uses Rezk’s K (1) -local logarithm ℓ : gl K → K [Rez06], togetherwith nontrivial knowledge of low-degree k -invariants for gl K . In rough, the factthat Rezk’s logarithm gives us an equivalence in degrees greater than implies thatthere is a diagram of fiber sequences: Σ ( ku ) ∧ p / / τ ≥ gl K / / (cid:15) (cid:15) gl K (cid:15) (cid:15) Σ Z p e e Z × p c c Applying [ Z /p, − ] gives us a spectral sequence that computes [ Z /p, Σ gl K ] ; in thecritical group the k -invariants of gl K give this spectral sequence one nontrivialdifferential for p > and two nontrivial differentials for p = 2 .However, the impossibility of adjoining these radicals can be shown directly using K (1) -local power operations, in a manner exactly analogous to the proof that onecannot adjoin roots of unity; in this form it generalizes. Let us sketch this argument.If we had such an E ∞ -ring K -algebra L , it would be p -complete and thus K (1) -local. Its coefficient ring K ∗ [ x ] / ( x p − α ) would then have a K (1) -local power operation ψ p , a ring endomorphism that agrees with the Frobenius mod p [Hop14]. Theelement ζ = ψ p ( x ) /x would then be a p ’th root of unity. If ζ is not in Z × p , then L is a K (1) -local E ∞ -ring containing a nontrivial p ’th root of unity and Devalapurkarhas shown this to be impossible [Dev17]. If ζ is in Z × p , then α ≡ ψ p x = ζ − x mod p ,which contradicts the fact that , x, . . . , x p − are a basis of this ring mod p . One source of formal roots is the theory of strictly commutative elements.
Definition 8.
The space G m ( R ) of strictly commutative units of R is the space ofmaps Z → gl ( R ) , or equivalently the space of E ∞ -maps Z → GL ( R ) . The genera-tor ∈ Z induces forgetful maps G m ( R ) → GL ( R ) → π gl ( R ) .In particular, a strictly commutative unit of R has an underlying unit in π ( R ) .6 roposition 9. Suppose α is a strictly commutative unit of R . Then, for any n > , α has a canonical lift to a formal n ’th root.Proof. The canonical Bockstein map Z /n → Σ Z can be composed with the map Z → gl ( R ) classifying α . Example . Let R = S − ( τ ≤ S ) be the localization of the first Postnikov truncationof the sphere spectrum with respect to some set S of primes (not containing ). Thenthere is a fiber sequence gl ( R ) → ( S − Z ) × k −→ Σ Z / . This k -invariant corresponds to a (2-torsion) homomorphism ( S − Z ) × → Z / . Thishomomorphism is a classical calculation of orientation theory: it is the map n if n ≡ +1 mod 4 , − if n ≡ − . As a result, one can determine the homotopy groups of the space of strictly commut-ing elements, and in particular there is an exact sequence → [ Z , gl ( R )] → [ Z , ( S − Z ) × ] → [ Z , Σ Z / . We find that any unit in S − Z which is congruent to mod lifts, essentiallyuniquely, to a strictly commutative unit of R . This allows us to construct commuta-tive algebras such as R [ √ and R [ √− , even though these are ramified extensionson the level of coefficient rings. Remark . In the case of strictly commutative units, we obtain a second descriptionof the algebra obtained by adjoining this root ρ . A strictly commutative elementdetermines a composite map S [ Z ] → S [GL ( R )] → R, and so we can construct the algebra R [ ρ ] as R ⊗ S [ Z ] S [ n Z ] .This has the benefit that it readily lifts to a nonunit version. If we define the strictly commutative multiplicative monoid M m ( R ) to be the space of E ∞ -maps N → M ( R ) = Ω ∞⊗ R to the multiplicative monoid of R , then a strictly commutative element α can havean n ’th root adjoined via the construction S [ n N ] ⊗ S [ N ] R. Remark . A further property possessed by strict units is that it is possible totrivialize them. Using the natural augmentation S [ Z ] → S of the spherical group7lgebra, any strictly commutative unit of R with underlying unit x ∈ π ( R ) gives riseto a new commutative ring spectrum R/ ( x −
1) = S ⊗ S [ Z ] R, whose underlying R -module is equivalent to the cofiber of the map R → R inducedby ( x − ∈ π ( R ) . This algebra has the following property: it is the universalcommutative R -algebra with a chosen commuting diagram Z / / (cid:15) (cid:15) gl ( R ) (cid:15) (cid:15) ∗ / / gl ( R/ ( x − . We will see similarly themed universal properties in later sections.
The shift-by-1 in our definition of formal roots is strongly suggestive: the suspendedunit spectrum
Σ gl ( R ) is a connective cover of the Picard spectrum pic ( R ) . In thissection we will begin exploring Picard-graded analogues of our constructions. Definition 13.
Suppose that A is an abelian group and C is a symmetric monoidal ∞ -category. The space of strict A -gradings for C is the space of maps A → pic ( C ) , orequivalently the space of E ∞ -maps A → Pic( C ) . There is a composite A → π pic ( C ) ,which we refer to as the underlying A -grading . Remark . Suppose that we have any symmetric monoidal functor A → C . Thespace A could be regarded as a discrete groupoid, so its image lies in C ≃ ; the objectsin A have inverses under the monoidal product, so monoidality of ρ implies thatits image lies inside Pic( C ) . We will not distinguish between symmetric monoidalfunctors A → C and symmetric monoidal functors A → Pic( C ) . Example . Let C be a symmetric monoidal ∞ -category. The strict Picard space of C , denoted by P ic( C ) , is the space of strict Z -gradings: maps Z → pic ( C ) , orequivalently E ∞ -maps Z → Pic( C ) . The generator ∈ Z induces forgetful maps P ic( C ) → Pic( C ) → π Pic( C ) , and we refer to the image as the underlying object . Example . The space of strict n -torsion objects of C , denoted by P ic [ n ] ( C ) , is thespace of strict Z /n -gradings: maps ρ : Z /n → pic ( C ) , or equivalently maps Z /n → Pic( C ) of E ∞ -spaces. The cofiber sequence Z n −→ Z → Z /n of spectra gives rise tothe following maps, where each double composite is a fiber sequence: µ n ( R ) → G m ( R ) n −→ G m ( R ) ∂ −→ P ic [ n ] ( R ) → P ic( R ) n −→ P ic( R ) In particular, the map ∂ sends a strictly commutative element α : Z → gl ( R ) to theimage Z /n → Σ Z α −→ Σ gl ( R ) → pic ( R ) of the formal n ’th root associated to α . 8 emark . For a commutative ring spectrum R , a strict n -torsion R -module withunderlying left R -module L has a choice of equivalence L ⊗ R n → R . If the module L is equivalent to R , then the map Z /n → π pic ( R ) is trivial and so the map lifts toa map Z /n → Σ gl ( R ) : a formal n ’th root. We can detect which root (up to n ’thpowers) by making a choice of an equivalence R → L ; this determines a compositeequivalence R ≃ R ⊗ R n → I ⊗ R n → R , and hence a unit in π ( R ) . Example . Suppose that C is a symmetric monoidal stable ∞ -category such that n is a unit in the ring π End C ( I ) . Then for k > there are isomorphisms π k pic ( C ) (cid:27) π k +1 Aut C ( I ) (cid:27) π k +1 End C ( I ) , and the latter are acted on invertibly by n . Therefore, the fiber sequence τ ≥ pic ( C ) → pic ( C ) → pic ( h C ) induces an equivalence Map( Z /n, pic ( C )) → Map( Z /n, pic ( h C )) . In this case, strict n -torsion objects in C are equivalent to strict n -torsion objects in the homotopycategory h C . Example . For a local ring k , there is a fiber sequence pic ( k ) → Z → Σ k × , where Σ k is a generator of π pic ( k ) . The k -invariant is the twist permutation of Σ k , and is represented by the homomorphism Z ։ {± } → k × . This k -invariantbecomes trivial on Z , and so the category of k -modules has a strict Z -grading.The spaces G m ( k ) are connected but not contractible, so the Z -gradings are uniqueup to isomorphism but not canonical. We can make them canonical by choosing a Z -grading of Z . Example
20 ([Law18, 1.3.7]) . For any commutative ring spectrum R , the element Σ R in Pic( R ) determines a map Z = π pic ( S ) → π pic ( R ) . For any d > , we get acomposite d Z → π pic ( R ) . If this lifts to a strict d Z -grading, we could call this an E d ∞ -structure on R : it should be a strengthening of the notion of an H d ∞ -structurefrom [BMMS86]. However, our enthusiasm for extending this notational conventionis very low.The universal property of the real bordism spectrum MO is that it is initialamong commutative ring spectra with a nullhomotopy of the map BO → Pic( S ) → Pic( MO ) of E ∞ -spaces. Equivalently, it is initial among commutative rings with acommutative diagram of spectra ko / / (cid:15) (cid:15) pic ( S ) (cid:15) (cid:15) Z / / pic ( MO ) . If in k , this k -invariant is trivial and so the category of k -modules has a strict Z -grading.Moreover, pic ( F ) ≃ Z , and so this Z -grading is canonical.
9n particular, this gives the real bordism spectrum MO a strict Z -grading and anycommutative MO -algebra inherits it. Similar considerations with complex or spinstructures structures give the complex bordism spectrum MU a strict Z -gradingand the spin bordism spectrum MSpin a strict Z -grading. Example . The Atiyah–Bott–Shapiro orientation lifts to give the complex K -theoryspectrum ku a commutative MU -algebra structure, and the real K -theory spectrum ko a commutative MSpin -algebra structure, due to work of Joachim [Joa04]. There-fore, ku admits a strict Z -grading and ko admits a strict Z -grading. Example . Let K C be the algebraic K -theory spectrum of the complex numbers,which comes equipped with a map f : K C → ku to the complex topological K -theory spectrum. Work of Suslin showed that the map f is an equivalence afterprofinite completion, and hence the fiber of f is rational. We would like to showthat K C has a strict Z -grading. (A similar argument applies to show that the strict Z -grading of the real K -theory spectrum ko lifts to the algebraic K -theory K R .)Consider the arithmetic square: K C / / (cid:15) (cid:15) K C Q (cid:15) (cid:15) K C ∧ / / ( K C ∧ ) Q The functor GL preserves this pullback. When we apply pic we get a diagram ofconnective spectra: pic ( K C ) / / (cid:15) (cid:15) pic ( K C Q ) (cid:15) (cid:15) pic ( K C ∧ ) / / pic (( K C ∧ ) Q ) On π this is the constant square Z , and on π we get a bicartesian square Z × / / (cid:15) (cid:15) Q × (cid:15) (cid:15) b Z × / / ( b Z Q ) × . Together these show that the diagram of Picard spectra is a homotopy pullbackdiagram. Let C be the cofiber of pic ( K C ) → pic ( K C ∧ ) ; its homotopy groups arethen rational above degree , and equal to the torsion-free group b Z × / Z × in degreeone.The obstruction to lifting the strict Z -grading Z → pic ( ku ) → pic ( ku ∧ ) ≃ pic ( K C ∧ ) to a Z -grading of pic ( K C ) is the map Z → C . However, consider the fibersequence τ ≥ C → C → Σ b Z × / Z × . Z into either term, andhence [ Z , C ] = 0 . Therefore, the strict Z -grading of ku can be extended to K C . The ring spectrum constructed by adjoining formal radicals turns out to be a specialcase of a more general construction associated to strict gradings. From this pointforward we will need to make heavier use of [Lur09, Lur17].
Definition 23.
Suppose that C is a presentable symmetric monoidal ∞ -category, A is an abelian group regarded as a discrete symmetric monoidal category, and that ρ : A → C is a strict A -grading. We define the trivializing algebra T ρ to be thehomotopy colimit of ρ . Remark . Since A is a discrete category, there is an equivalence in C of the form T ρ ≃ a a ∈ A ρ ( a ) . This homotopy colimit is a very special case of the Thom construction. As such,work of Antolín-Camarena–Barthel gives the trivializing algebra a universal property,generalizing the results of [May77, ABG + ] from the category of spectra. Proposition 25.
Suppose that A is an abelian group and that ρ : A → C is a symmetricmonoidal functor with trivializing algebra T ρ .1. T ρ has a natural lift to a commutative algebra object: T ρ ∈ CAlg( C ) .2. The algebra T ρ is universal among commutative algebras in CAlg( C ) with a chosencommuting diagram A ρ / / (cid:15) (cid:15) pic ( C ) (cid:15) (cid:15) ∗ / / pic (LMod T ρ ) . In particular, for any a ∈ A the algebra T ρ has a chosen equivalence of left T ρ -modules φ a : T ρ → T ρ ⊗ ρ ( a ) , and there are chosen coherences φ a ⊗ ◦ φ b ≃ φ ab .Proof. The colimit has a commutative algebra structure by [AB14, Theorem 2.8].We will now prove the universal property essentially, following the same lineof argument in [AB14, Lemma 3.15]. Applying [AB14, Theorem 2.13] to the functor A → C of symmetric monoidal ∞ -categories, we find the following: maps T ρ → R in CAlg( C ) are equivalent to lax symmetric monoidal lifts in the diagram C /R (cid:15) (cid:15) A / / Pic( C ) / / C . C /R are maps N → R , with symmetric monoidal product given by ( N → R ) ⊗ ( M → R ) ≃ ( N ⊗ M → R ⊗ R → R ) . The monoidal unit is the map I → R . There is a symmetric monoidal functor C /R → C , given by forgetting the structure map, and a symmetric monoidal functor C /R → (LMod R ) /R , given by ( L → R ) ( R ⊗ L → R ) .However, since A is grouplike the image L → R of any object must be containedin the invertible objects of C /R . This implies first that L is an invertible object of C . This implies second that R ⊗ L → R is an invertible object of (LMod R ) /R ; thishappens only when this adjoint structure map is an equivalence. Conversely, if L → R is a map whose adjoint R ⊗ L → R is an equivalence, tensoring with L − gives an equivalence of R -modules R → R ⊗ L − of left R -modules, whose inverse isadjoint to a map L − → R . Remark . Suppose that the functor ρ : A → C maps entirely to the unit compo-nent: it is a map A → B Aut C ( I ) of E ∞ -spaces, classifying an extension G of theautomorphism space Aut C ( I ) . Then the trivializing algebra T ρ is universal amongcommutative algebras R in C where the map Aut C ( I ) → Aut C ( R ) extends to a mapof E ∞ -spaces G → Aut C ( R ) .In particular, when C is the category Mod R of modules over a commutative ringspectrum, this universal property recovers the relative tensor product defined earlier. Example . The trivializing algebra for the map Z → pic ( MU ) is a periodic MU -spectrum MUP , whose coefficient ring is π ∗ MU [ u ± ] for a generator u in degree .It is universal among commutative algebras R with a nullhomotopy of the composite ku → pic ( S ) → pic ( R ) . The algebra MUP is often useful for translating betweeneven-periodic and Z / -graded interpretations in chromatic theory. The symmetric monoidal functor from C to its homotopy category h C induces a mapof Picard spectra pic ( C ) → pic ( h C ) , identifying pic ( h C ) with the first nontrivial stage τ ≤ pic ( C ) in the Postnikov tower for pic ( C ) . For us to lift a map ¯ ρ : A → π pic ( C ) to the first Postnikov stage pic ( h C ) , it is necessary and sufficient that the compositemap A ¯ ρ −→ π pic ( C ) k −→ Σ π pic ( C ) is trivial. The result is an obstruction class: an element of [ A, Σ π pic ( C ))] , and alift exists if and only if this obstruction vanishes. Two different choices of lift to amap A → pic ( h C ) are represented by homotopy classes of maps A → Σ π pic ( C ) .The identification of the first k -invariant with the twist homomorphism τ leadsus to the following definition [Rez09]. Definition 28.
An element γ ∈ π pic ( C ) is symmetric if τ ( γ ) = id in π pic ( C ) .This allows us to conclude the following.12 roposition 29. A map ¯ ρ : A → π pic ( C ) lifts to a map A → pic ( h C ) if and only ifthe elements ¯ ρ ( a ) are symmetric for all a ∈ A . Two such lifts determine a difference classin Ext(
A, π pic ( h C )) .Example . Suppose that L is an invertible object such that there is an equivalence v : I → L ⊗ n . The object L determines a map S → pic ( C ) , and the equivalence v determines a nullhomotopy of n times this map. Alternatively, there is a commutativediagram of symmetric monoidal ∞ -categories F ( L ⊗ n ) / / (cid:15) (cid:15) F ( L ) (cid:15) (cid:15) ∗ / / Pic( C ) , where F ( x ) is the free E ∞ -space on an object named x . Taking associated spectragives a diagram S n / / (cid:15) (cid:15) S (cid:15) (cid:15) ∗ / / pic ( C ) , or equivalently a map S /n → pic ( C ) . Conversely, there is a short exact sequence → π pic ( C ) / [ π pic ( C )] n → [ S /n, pic ( C )] → π pic ( C )[ n ] → . The quotient expresses that a map from S /n determines an underlying n -torsionobject L in π pic ( C ) . The kernel expresses that two different maps representingthe same object L may differ in their choice of equivalence v : I → L ⊗ n (moduloself-equivalences of L ).If there is a symmetric monoidal functor f : C → D such that there is an equiv-alence u : I → f ( L ) in D , then the map S → pic ( D ) determining f ( L ) becomestrivial. However, the extended map S /n → pic ( D ) does not always become trivial: itbecomes trivial precisely when there exists a choice of u : I → f ( L ) in D such that u n = f ( v ) .The bottom homotopy group of S /n is Z /n . The map S /n → pic ( h C ) extends toa map Z /n → pic ( h C ) if and only if L is symmetric, and if so it extends uniquely. Astrict Z /n -grading would be an extension to a map Z /n → pic ( C ) . In rough, we canthink of this in the following way. If we have a strict Z /n -grading, then it is a rigidversion of choosing an object L with an equivalence v : I → L ⊗ n ; the trivializingalgebra T then extracts an n ’th root of this chosen equivalence v . Example . Suppose that R is a commutative ring spectrum which is n -periodic:there is a unit v ∈ π n R . This determines a symmetric element Σ R , and a lift of Σ R to a strict n -torsion object allows us to construct an R -algebra whose Z -gradedcoefficient ring is π ∗ R [ x ] / ( x n − av ) a ∈ ( π R ) × / [( π R ) × ] n . If n is a unit in π R then we also findthat such algebras can be constructed, essentially uniquely, for any value of a . Theseextensions appear, for example, when relating completed Johnson–Wilson spectra toLubin–Tate spectra [LN12, §4]. In this section, we will begin the process of extending gradings by adjoining formalradicals to elements in the Picard group. We fix a symmetric monoidal presentable ∞ -category C , and let Pic( C ) be the Picard space of invertible elements in C . Definition 32.
Let → π pic ( C ) → Γ → B → be an extension of abelian groups, represented by a map ¯ ρ : B → Σ π pic ( C ) between Eilenberg–Mac Lane spectra. A extension to Γ -grading is a lift of ¯ ρ to pic ( C ) : Σ pic ( C ) (cid:15) (cid:15) B ρ ; ; ✈✈✈✈✈✈✈✈✈✈ ¯ ρ / / Σ π pic ( C ) We refer to the fiber of ρ as the extended Picard spectrum pic ( C , ρ ) associated to ρ ,and the associated infinite loop space as the extended Picard group Pic( C , ρ ) .The extended Picard spectrum is part of a fiber sequence pic ( C ) → pic ( C , ρ ) → B, and on π this realizes the extension → π pic ( C ) → Γ → B → . We get adecomposition Pic( C , ρ ) (cid:27) ` b ∈ B Pic( C ) · { b } as spaces with an action of Pic( C ) . Remark . Again, the map ρ determines ¯ ρ and the extension Γ .By definition, there is an inclusion i : Pic( C ) ⊂ C of symmetric monoidal ∞ -categories. The latter is presentable, whereas the former is (essentially) small. Byformally adjoining colimits to C , we obtain a factorization Pic( C ) → P (Pic( C )) → C through the presheaf ∞ -category, where the second functor preserves colimits. Proposition 34.
For a symmetric monoidal ∞ -category C , there is a diagram P (Pic( C , ρ )) ← P (Pic( C )) → C of symmetric monoidal presentable ∞ -categories. roof. Fix a regular cardinal κ such that the unit of C is κ -compact. Then all ofthe objects of Pic( C ) are contained inside the essentially small subcategory C κ of κ -compact objects.The functor P is the left adjoint in an adjunction between κ -small ∞ -categoriesand κ -presentable ∞ -categories; ( − ) κ is the right adjoint. Moreover, the tensorproduct of presentable ∞ -categories is universal with respect to functors that arecolimit-preserving in each variable separately; in particular, this gives us canonicalidentifications Fun
PrL ( P ( Π S i ) , C ) ≃ Fun
PrL ( ⊗P ( S i ) , C ) natural in C . This makes the functor P strong symmetric monoidal, and lifts it to aleft adjoint to the functor taking a κ -presentable symmetric monoidal ∞ -category tothe symmetric monoidal subcategory of κ -compact objects. For a small symmetricmonoidal ∞ -category S , the induced symmetric monoidal structure on the category P ( S ) is given by left Kan extension: this is the Day convolution monoidal structure[Gla16, Lur17]. It is colimit-preserving in each variable, and for objects s and t of S with associated presheaves j s and j t there is a natural isomorphism j s ⊗ j t (cid:27) j s ⊗ t .The Day convolution makes the functor P (Pic( C )) → P (Pic( C , ρ )) symmetricmonoidal, and the adjunction gives us a composite symmetric monoidal functor P (Pic( C )) → P ( C κ ) → C as desired. Definition 35.
Suppose that ρ is an extension to Γ -grading. We define the categoryobtained by extending gradings to Γ to be the symmetric presentable ∞ -category C [ ρ ] = P (Pic( C , ρ )) ⊗ P (Pic( R )) C . Proposition 36.
As a presentable category left-tensored over C , we have C [ ρ ] (cid:27) Y b ∈ B C . In particular, C [ ρ ] is isomorphic to the category of B -graded objects of C .Proof. Since
Pic( C , ρ ) (cid:27) ` b ∈ B Pic( C ) as categories left-tensored over Pic( C ) , P (Pic( C , ρ )) (cid:27) pres a b ∈ B P (Pic( C )) as presentable ∞ -categories left-tensored over P ( C ) —here the coproduct takingplace within presentable ∞ -categories. The relative tensor product preserves col-imits in each variable, and thus we have C [ ρ ] ≃ pres a b ∈ B C as categories left-tensored over C . However, within presentable ∞ -categories, co-products and products over a small index set coincide.15ne source of grading extensions is the theory of strict gradings. Proposition 37.
Suppose that → G → Γ → B → is an extension of abelian groups.Then every strict G -grading of C has a naturally associated extension to a Γ -grading.Proof. The extension Γ is classified by a map B → Σ G , which can be composed withthe strict G -grading G → pic ( C ) . Example . Since MO has a strict Z -grading, for any MO -algebra R we can thenadjoin invertible objects L to the category of MO -modules such that L ⊗ n ≃ Σ MO ,or extend to any grading Z ⊂ Γ . For example, we can embed the category of MO -modules into the category of Q / Z -graded MO -modules, giving the latter asymmetric monoidal structure where shifts by integers are extended to shifts byrational numbers. Example . Similarly, the strict Z -grading on MU allow us to adjoin invertibleobjects L such that L ⊗ n (cid:27) Σ MU . (Note that if we take n = 2 in this construction, wefind that the object Σ − L is a nontrivial object with ( Σ − L ) ⊗ (cid:27) MU .) This allows usto extend from a Z -grading on MU -modules to a grading over Z ⊕ Z Q (cid:27) Q × Z / .Similar constructions are possible with MU -algebras like ku or with the algebraic K -theory spectrum K C . As in §6, we can identify pic ( h C ) with the first nontrivial stage τ ≤ pic ( C ) in thePostnikov tower for pic ( C ) . For us to lift a map ¯ ρ : B → Σ π pic ( C ) to the firstPostnikov stage, it is necessary and sufficient that the associated obstruction B ¯ ρ −→ Σ π pic ( C ) k −→ Σ π pic ( C ) is trivial. Two different choices of lift to a map B → Σ pic ( h C ) differ by an elementof [ B → Σ π pic ( C )] .This can be concisely packaged into the following result. Proposition 40.
Given an extension → π pic ( C ) → Γ → B → , the lifts of the map ¯ ρ : B → Σ π pic ( C ) to a map ρ ≤ : B → Σ pic ( h C ) are in bijective correspondence withextensions of the twist homomorphism τ : π pic ( C ) → π pic ( C )[2] to all of Γ .Example . This construction can recover the twisted Z / -graded categories of Rezk[Rez09, §2]. Let C be an ordinary presentable symmetric monoidal category and ω an invertible object of C . We would like to construct a larger category C [ √ ω ] where ω is the square of another invertible module. This would be a symmetric monoidalcategory of Z / -graded objects ( A , A ) of C , representing A ⊕ ( √ ω ⊗ A ) , with atensor product satisfying ( A , A ) ⊗ ( B , B ) (cid:27) (cid:18) ( A ⊗ B ) ⊕ ( ω ⊗ A ⊗ B ) , ( A ⊗ B ) ⊕ ( A ⊗ B ) (cid:19) . Rezk gives these constructions in the case where C is merely additive, which is not covered by ourassumption that C is presentable.
16n this case, the underlying category is ordinary and so pic ( C ) = pic ( h C ) . For usto extend gradings, it is necessary and sufficient that ω be symmetric. If this isthe case, the possible choices of extension represent choices of 2-torsion element τ ( √ ω ) ∈ Aut C ( I ) , a Koszul sign rule for the twist isomorphism on the square rootof ω . If C is additive, choosing τ ( √ ω ) = − recovers Rezk’s ω -twisted tensor product . Just as the unit spectrum is extended by the Picard spectrum, the Picard spectrumis extended by the Brauer spectrum [GL16, Hau17]. Fix a symmetric monoidal pre-sentable ∞ -category C and let Cat C be the category of presentable ∞ -categoriesleft-tensored over C . This has a symmetric monoidal under ⊗ C . We write Br( C ) =Pic(Cat C ) for the Brauer space parametrizing invertible objects of Cat C that admita compact generator, and br ( C ) for the associated spectrum.The unit of Cat C is C itself, and all C -linear functors are of the form X X ⊗ B for some B ∈ C ; in particular, this identifies the C -linear functors C → C with C ≃ itself,with composition given by the tensor in C . As a result, the space of self-equivalencesof the unit is Pic( C ) , and so there is a fiber sequence Σ pic ( C ) → br ( C ) → π br ( C ) . Definition 42.
Suppose that C is a presentable symmetric monoidal ∞ -category, B is an abelian group regarded as a discrete symmetric monoidal category, and that ρ : B → Cat( C ) is a symmetric monoidal functor. We define the trivializing category T ρ to be the homotopy colimit of ρ , calculated in Cat C . Remark . Since B is a discrete category, there is an equivalence in Cat C of theform T ρ ≃ Cat C a b ∈ B ρ ( b ) ≃ Y b ∈ B ρ ( b ) . Proposition 44.
Suppose that B is an abelian group and that ρ : B → C is a symmetricmonoidal functor with trivializing category T ρ .1. The map ρ has a natural lift to a map B → Br( C ) .2. T ρ has a natural lift to a symmetric monoidal ∞ -category under C .3. The symmetric monoidal ∞ -category T ρ is universal among symmetric monoidalpresentable ∞ -categories under C with a chosen commutative diagram B / / (cid:15) (cid:15) br ( C ) (cid:15) (cid:15) ∗ / / br ( T ρ ) . n particular, for any b ∈ B there is an equivalence φ b : T ρ → T ρ ⊗ C ρ ( b ) ofpresentable ∞ -categories left-tensored over T ρ , and there are chosen equivalences offunctors φ a ⊗ ◦ φ b ∼ −→ φ ab .Proof. This is Proposition 25, applied to the (large) ∞ -category of presentable ∞ -categories. Example . Let C = LMod R where R is a commutative ring spectrum. Then eachobject of Br( R ) is represented by the category of right modules over an Azumaya R -algebra Q which is well-defined up to Morita equivalence [AG14, Toë12, BRS],see also [GL16, 5.13]. Given a symmetric monoidal functor ρ : B → Br( R ) , we cantherefore choose algebras Q ( b ) so that there is an equivalence T ρ ≃ Y b ∈ B RMod Q ( b ) . The symmetric monoidal structure on this category takes more work to describe. Thesymmetric monoidal structure on ρ gives Morita equivalences between Q ( p ) ⊗ R Q ( q ) and Q ( p + q ) , which are expressed by Q ( p ) ⊗ R Q ( q ) ⊗ R Q ( p + q ) op -modules M p,q . Thesymmetric monoidal structure is given by a formula of the form ( X b ) b ∈ B ⊗ ( Y b ) b ∈ B (cid:27) M p + q = b ( X p ⊗ Y q ) ⊗ Q ( p ) ⊗ Q ( q ) M p,q b ∈ B . However, describing the full symmetric monoidal structure in this fashion wouldrequire us to carefully express coherence relations between tensor products of thebimodules M p,q . We can think of these objects as “coefficients for the multiplication”that give the symmetric monoidal structure on this product category. Remark . The first k -invariant in the Brauer spectrum is a map π br ( C ) → Σ π pic ( C ) , and this is still expressed by the twist self-equivalence. However, nowthis twist self-equivalence occurs on the level of module categories. For a com-mutative ring spectrum R with an Azumaya R -algebra Q , the twist equivalence of RMod Q ⊗ R Q is expressed by tensoring with the Q ⊗ R Q -bimodule Q ⊗ R Q τ , wherethe action on the left is the standard one and on the right factors through the twistautomorphism. The fact that Q ⊗ R Q is Azumaya means that this bimodule must beof the form Q ⊗ R Q ⊗ R τ ( Q ) for some τ ( Q ) ∈ Pic( R ) . More concretely, we can identify τ ( Q ) with the R -module F Q ⊗ R Q - bimod ( Q ⊗ R Q, ( Q ⊗ R Q ) τ ) . This assignment remains relatively mysterious to the author. In algebraic examplesit is trivial, which is shown by descent-theoretic methods in [Vis]. The algebraic version of this assignment, taking k -algebra automorphisms of an Azumaya algebra T to elements in the Picard group, is part of the Rosenberg–Zelinsky exact sequence → k × → T × → Aut
Alg( k ) ( T ) → π pic ( k ) that expresses the potential failure of algebra automorphisms to be inner. If k is a field this recovers theNoether–Skolem theorem. K -theory We close with some conjectural remarks on the relationship with algebraic K -theory.We recall the following from [GL16, 5.11]. The functor Mod is symmetric monoidalby [Lur17, 4.8.5.16]: for A and B R -algebras, the tensor product in
Mod R -linear ∞ -categories has a natural equivalence Mod A ⊗ Mod B ≃ Mod A ⊗ R B . Moreover, by [GGN15, 8.6] the algebraic K -theory functor is lax symmetric monoidal,and hence induces a functor Cat R → Mod KR . While this functor is only lax symmetric monoidal, it does send the unit
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