Commutative algebras

In this section we given an easier description of the cyclotomic structure on THH(A) for anE∞-ring spectrumA. First, we show that, ifAis anE∞-ring spectrum, then THH(A) is again anE∞-ring spectrum. This follows from the following construction.

Construction IV.2.1. The∞-categories Cyc Sp and Cyc Sppof (p-)cyclotomic spec-tra have a natural symmetric monoidal structure. This follows from the following two observations:

(i) If C is a symmetric monoidal ∞-category, then C^S=Fun(S,C) is naturally a symmetric monoidal ∞-category for any simplicial set S. Indeed, one can define the total space as Fun(S,C)^⊗=Fun(S,C^⊗)×Fun(S,N(Fin∗))N(Fin∗), and it is easy to verify that this defines a symmetric monoidal ∞-category, since exponentials of cocartesian fibrations are again cocartesian [69, Corollary 3.2.2.12].

(ii) If C and D are symmetric monoidal ∞-categories, F:C!D is a symmetric monoidal functor and G:C!D is a lax symmetric monoidal functor, then LEq(F, G) has a natural structure as a symmetric monoidal∞-category. Indeed, one can define the total space as the pullback

LEq(F, G)^⊗ //

(D^⊗)^∆_id¹

C^{⊗ (F}

⊗,G^⊗) //D^⊗×D^⊗,

where (D^⊗)^∆_id¹⊆(D^⊗)^∆1 is the full subcategory consisting of those morphisms which lie over identities inNFin^∆_∗¹, and whereF^⊗, G^⊗:C^⊗!D^⊗ are the functors on total spaces.

In particular, LEq(F, G)^⊗ is a full subcategory of LEq(F^⊗, G^⊗), and for any symmetric monoidal∞-category E, giving a symmetric monoidal (resp. lax symmetric monoidal) functorE!LEq(F, G) is equivalent to giving a symmetric monoidal (resp. lax symmet-ric monoidal) functorH:E!C together with a lax symmetric monoidal transformation FH!GH.

Moreover, Alg_E

1(Sp) is naturally a symmetric monoidal∞-category. It follows from the discussion of lax symmetric monoidal structures on all intervening objects in§§III.1–

III.3that the functor

THH: Alg_E1(Sp)−!Cyc Sp

is naturally a lax symmetric monoidal functor. In fact, it is symmetric monoidal as this can be checked on the underlying spectrum, where one uses that geometric realizations (or more generally sifted colimits) commute with tensor products in a presentably symmetric

monoidal∞-category. In particular, THH mapsE∞-algebras toE∞-algebras. Moreover, we recall that

Alg_E∞(Sp) = Alg_E∞(Alg_E

1(Sp)),

so indeed THH(A) is an E∞-algebra in cyclotomic spectra if Ais an E∞-algebra. Con-cretely, this means that THH(A) is aT-equivariantE∞-algebra in spectra together with aT/C_p∼=T-equivariant mapϕ_p: THH(A)!THH(A)^tCp ofE∞-algebras.

Moreover, we note that the inclusion of the bottom cells in the cyclic objects define a commutative diagram

A //

∆_p

THH(A)

ϕp

(A⊗...⊗A)^tCp //THH(A)^tCp. In fact, one has a lax symmetric monoidal functor from Alg_E

1(Sp) to the symmetric monoidal ∞-category of such diagrams. In particular, if A is an E∞-algebra, then all maps areE∞-ring maps.

Recall the following fact.

Proposition IV.2.2. (McClure–Schw¨anzl–Vogt [79]) For an E∞-ring spectrum A, the map A!THH(A)is initial among all maps from Ato anE∞-ring spectrum equipped with a T-action (through E∞-maps).

Proof. We have to prove that THH(A) is the tensor ofAwithS¹ in the∞-category ofE∞-ring spectra. We use the simplicial model for the circleS¹ given asS=∆¹/∂∆¹ which has n+1 different (possible degenerate) n-vertices Sn. Thus, we have in the ∞-category S of spaces the colimit S¹'colim∆^opSn. Therefore, we get that the tensor of RwithS¹ is given by

colim∆^opR^⊗Sn'THH(R),

where we have used thatR^⊗Sn=R^⊗(n+1)is the (n+1)-fold coproduct in the∞-category ofE∞-ring spectra; cf. [71, Proposition 3.2.4.7].

From theE∞-map A!THH(A), we get aC_p-equivariantE∞-map A⊗...⊗A−!THH(A)

by taking the coproduct in the category ofE∞-algebras of all the translates by elements of C_p⊆T of the map (note that A⊗...⊗A is the inducedC_p-object in E∞-rings). This

is the map that can also be described through the p-fold subdivision. Thus, in the commutative square

A //

∆p

THH(A)

ϕ_p

(A⊗...⊗A)^tCp //THH(A)^tCp

(14)

ofE∞-rings, the lower map is explicit, giving rise to an explicit map A!THH(A)^tCp of E∞-rings, where T=T/C_p acts on THH(A)^tCp. In view of the above universal property of THH(A), we conclude the following.

CorollaryIV.2.3. For anE∞-ringA,the Frobeniusϕ_pis the uniqueT-equivariant E∞-map THH(A)!THH(A)^tCp that makes the diagram (14)commutative.

Note that this can also be used to define ϕ_p in this situation. Moreover, this obser-vation can be used to prove that the various definitions ofE∞-ring structures on THH(A) in the literature are equivalent to ours.

A consequence of PropositionIV.2.2is that, for E∞-ring spectraA, there is always a map π: THH(A)!A which is a retract of the map A!THH(A). The map π is, by construction,T-equivariant whenAis equipped with the trivial T-action, in contrast to the map in the other direction.

Corollary IV.2.4. For anE∞-ringAthe composition A−!THH(A)−−−^ϕp!THH(A)^tCp−−−−^πtCp!A^tCp is equivalent to the Tate-valued Frobenius (see Definition IV.1.1).

Proof. We have a commutative diagram

A //

∆_p

THH(A)

ϕp

(A⊗...⊗A)^tCp //THH(A)^{tCp πtCp} //A^tCp.

(15)

Therefore, it suffices to show that the composition A⊗...⊗A−!THH(A)−!A

is as aC_p-equivariant map equivalent to the multiplication of A. The source is, as a C_p-object inE∞-algebras, induced fromA. Therefore, this amounts to checking that the mapA!THH(A)!Ais equivalent to the identity, which is true by definition.

A slightly different perspective on the construction ofϕp in the commutative case is due to Jacob Lurie. To this end, letR=THH(A), or more generally letRbe anyE∞-ring spectrum withCp-action. Then, we have a map ofE∞-ring spectra which can informally be described as

m:e R⊗...⊗R−!R, r₁⊗...⊗r_p7−!

p

Y

i=1

σⁱ(r_i),

whereσ∈Cp is a generator. The precise way of defining this map is by the observation that the left-hand side is the induced E∞-ring spectrum with an action by Cp. Then, we get the desired map me by the universal property. This also shows that the map me is C_p-equivariant where the left-hand side is equipped with the cyclic action and the right-hand map with the given action. Now, consider the composite map

ϕep:R−−−^∆p!(R⊗...⊗R)^tCp−−−−−^fmtCp!R^tCp.

This is an equivariant version of the Frobenius map discussed above. Now, in the case where R=THH(A), we not only have aCp-action, but a T-action, and the mapϕep is, by functoriality,T-equivariant, where the target is equipped with theT-action obtained from the T-action on R that extends the Cp-action. This action ofT onR^tCp has the property thatCp⊆T acts trivially, so that theT-action factors over the residualT/Cp -action onR^tCp that we have used several times before. As a result, we get a factorization of the mapϕep through the homotopy orbits of theCp-action onR in the∞-category of E∞-ring spectra. Writing R=THH(A) as the tensor ofA with T in the∞-category of E∞-rings, these homotopy orbits are the tensor of A with T/Cp∼=T in the ∞-category ofE∞-rings. Thus, they are equivalent to THH(A) itself, in aT∼=T/Cp-equivariant way.

In total, we get aT-equivariant map ofE∞-rings

ϕ: THH(A)−!THH(A)^tCp.

It follows from the construction that it sits in a commutative diagram ofE∞-rings

A //

∆_p

THH(A)

ϕ

(A⊗...⊗A)^tCp //THH(A)^tCp, so it must by CorollaryIV.2.3agree withϕ_p.