Splittings and calculational techniques for higher THH
Irina Bobkova, Eva Höning, Ayelet Lindenstrauss, Kate Poirier, Birgit Richter, Inna Zakharevich
aa r X i v : . [ m a t h . A T ] N ov SPLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH
IRINA BOBKOVA, EVA H ¨ONING, AYELET LINDENSTRAUSS, KATE POIRIER, BIRGIT RICHTER,AND INNA ZAKHAREVICH
Abstract.
Tensoring finite pointed simplicial sets X with commutative ring spectra R yieldsimportant homology theories such as (higher) topological Hochschild homology and torushomology. We prove several structural properties of these constructions relating X ⊗ ( − ) toΣ X ⊗ ( − ) and we establish splitting results. This allows us, among other important examples,to determine THH [ n ] ∗ ( Z /p m ; Z /p ) for all n > m > Introduction
For any (finite) pointed simplicial set X one can define the tensor product of X with acommutative ring spectrum A , X ⊗ A , where the case X = S gives topological Hochschildhomology of A . More generally, for any sequence of maps of commutative ring spectra R → A → C we can define L RX ( A ; C ), the Loday construction with respect to X of A over R withcoefficients in C . Important examples are X = S n or X a torus. The construction specializesto X ⊗ A in the case L SX ( A ; A ). For details see Definition 1.1.An important question about the Loday construction concerns the dependence on X : Given X, Y ∈ s Set ∗ , with Σ X ≃ Σ Y , does that imply that L RX ( A ) ≃ L RY ( A )? If it does, the Lodayconstruction would be a “stable invariant”. Positive cases arise from the work of Berest,Ramadoss and Yeung [5, Theorem 5.2]: They identify the homotopy groups of the Lodayconstruction over an X ∈ s Set ∗ of a Hopf algebra over a field with representation homologyof the Hopf algebra with respect to Σ( X + ). In [16] Dundas and Tenti prove that stableinvariance holds if A is a smooth algebra over a commutative ring k . However in [16] theyalso provide a counterexample: L H Q S ∨ S ∨ S ( H Q [ t ] /t ) is not equivalent to L H Q S × S ( H Q [ t ] /t )even though Σ( S ∨ S ∨ S ) ≃ Σ( S × S ). Our juggling formula (Theorem 3.2) and ourgeneralized Brun splitting (Theorem 4.1) relate the Loday construction on Σ X to that of X . One application among others of these results is to establish stable invariance in certainexamples.For commutative F p -algebras A one often observes a splitting of THH ( A ) as THH ( F p ) ∧ H F p THH H F p ( HA ), so THH ( A ) splits as topological Hochschild homology of F p tensored with theHochschild homology of A [25]. It is natural to ask in which generality such splittings occur.If one replaces F p by Z , then there are many counterexamples. For instance if A = O K is anumber ring then THH ∗ ( O K ) is known by [27, Theorem 1.1] and is far from being equivalent Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary 18G60; Secondary 55P43.
Key words and phrases. higher topological Hochschild homology, higher Shukla homology. to π ∗ ( THH ( Z ) ∧ LH Z THH H Z ( H O K )) in general (see [19, Remark 4.12.] for a concrete example).We prove several splitting results for higher THH and use one of them to determine higher
THH of Z /p m with Z /p -coefficients for all m > Content.
We start with a brief recollection of the Loday construction in Section 1.In [6] we determined the higher Hochschild homology of R = F p [ x ] and of R = F p [ x ] /x p ℓ ,both Hopf algebras. In Section 2 we generalize our results to the cases R = F p [ x ] /x m if p divides m , which is not a Hopf algebra unless m = p ℓ for some ℓ .Building on work in [19] we prove a juggling formula (see Theorem 3.2): For every sequenceof cofibrations of commutative ring spectra S → R → A → B → C there is an equivalence L R Σ X ( B ; C ) ≃ L R Σ X ( A ; C ) ∧ L L AX ( C ) L BX ( C ) . We explain in 3.1 what happens if one oversimplifies this formula.Using a geometric argument, Brun [9] constructs a spectral sequence for calculating
THH ∗ -groups. We prove a generalization of his splitting and show that for any sequence of cofibra-tions of commutative ring spectra S → R → A → B we obtain a generalized spectrum-levelBrun splitting (see Theorem 4.1) i.e. , an equivalence of commutative B -algebra spectra L R Σ X ( A ; B ) ≃ B ∧ L L RX ( B ) L AX ( B ) . Note that B , which only appears at the basepoint on the left, now appears almost everywhereon the right. This splitting also gives rise to associated spectral sequences for calculatinghigher THH ∗ -groups.We apply our results to prove a generalization of Greenlees’ splitting formula [18, Remark7.2]: For an augmented commutative k -algebra A we obtain in Corollary 4.6 that L Σ X ( HA ; Hk ) ≃ L Σ X ( Hk ) ∧ LHk L HAX ( Hk )and if A is flat as a k -module then this can also be written as L Σ X ( HA ; Hk ) ≃ L Σ X ( Hk ) ∧ LHk L Hk Σ X ( HA ; Hk ) , where all the Loday constructions are over the same simplicial set. For X = S n , for example,and A a flat augmented commutative k -algebra this yields Theorem 5.10, THH [ n ] ( A ; k ) ≃ THH [ n ] ( k ) ∧ LHk
THH [ n ] ,k ( A ; k )where THH [ n ] = L S n .Shukla homology is a derived version of Hochschild homology. We define higher orderShukla homology in Section 5 and calculate some examples that will be used in subsequentresults. We prove that the Shukla homology of order n of a ground ring k over a flat aug-mented k -algebra is isomorphic to the reduced Hochschild homology of order n + 1 of the flataugmented algebra (Proposition 5.9).Tate shows [40] how to control Tor-groups for certain quotients of regular local rings. Weuse this to develop a splitting on the level of homotopy groups for THH ( R/ ( a , . . . , a r ); R/ m )if R is regular local, m is the maximal ideal and the a i ’s are in m . PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH We prove a splitting result for
THH [ n ] ( R/a, R/p ) in Section 7, where R is a commutativering and p, a ∈ R are not zero divisors, ( p ) is a maximal ideal, and a ∈ ( p ) . In this situation, THH [ n ] ( R/a, R/p ) ≃ THH [ n ] ( R, R/p ) ∧ LHR/p
THH [ n ]; R ( R/a, R/p ) . In many cases the homotopy groups of the factors on the right hand side can be completelydetermined. Among other important examples we get explicit formulas for
THH [ n ] ( Z /p m , Z /p )for all n > m > THH [ n ] ∗ ( Z /p m , Z /p ) ∼ = THH [ n ] ∗ ( Z , Z /p ) ⊗ Z /p THH [ n ] , Z ∗ ( Z /p m , Z /p ) . We know
THH [ n ] ∗ ( Z , Z /p ) from [13] and we determine THH [ n ] , Z ∗ ( Z /p m , Z /p ) explicitly for all n . This generalizes previous results by Pirashvili [34], Brun [9], and Angeltveit [1] from n = 1to all n .In Section 8 we provide a splitting result for commutative ring spectra of the form A × B :we show in Proposition 8.3 that for any finite connected simplicial set X , we have L X ( A × B ) ≃ / / L X ( A ) × L X ( B ) . We present some sample applications of our splitting results in Section 9: a splitting ofhigher
THH of ramified number rings with reduced coefficients (9.2), a version of Galoisdescent for higher
THH (9.3) and a calculation of higher
THH of function fields over F p (9.4).We close with a discussion of the case of higher THH of Z /p m (with unreduced coefficients)(9.5). Acknowledgements.
This paper grew out of a follow-up project to our
Women in TopologyI project on higher order topological Hochschild homology [6]. We were supported by an AIMSQuaRE grant that allowed us to meet in 2015, 2016 and 2018. AL acknowledges support bySimons Foundation grant 359565. BR thanks the Department of Mathematics of the IndianaUniversity Bloomington for invitations in 2016, 2017 and 2018. AL and BR thank MichaelLarsen for lessening their ignorance of algebraic number theory. We thank Bjørn Dundas,Mike Mandell, and Brooke Shipley for helpful comments.1.
The Loday construction: basic features
We recall some definitions concerning the Loday construction and we fix notation.For our work we can use any good symmetric monoidal category of spectra whose categoryof commutative monoids is Quillen equivalent to the category of E ∞ -ring spectra, such assymmetric spectra [23], orthogonal spectra [29] or S -modules [17]. As parts of the paperrequire to work with a specific model category we chose to work with the category of S -modules.Let X be a finite pointed simplicial set and let R → A → C be a sequence of maps ofcommutative ring spectra. BOBKOVA, H ¨ONING, LINDENSTRAUSS, POIRIER, RICHTER, AND ZAKHAREVICH
Definition 1.1.
The Loday construction with respect to X of A over R with coefficients in C is the simplicial commutative augmented C -algebra spectrum L RX ( A ; C ) whose p -simplicesare C ∧ ^ x ∈ X p \∗ A where the smash products are taken over R . Here, ∗ denotes the basepoint of X and we placea copy of C at the basepoint. As the smash product over R is the coproduct in the category ofcommutative R -algebra spectra, the simplicial structure is straightforward: Face maps d i on X induce multiplication in A or the A -action on C if the basepoint is involved. Degeneracies s i on X correspond to the insertion of the unit maps η A : R → A over all n -simplices whichare not hit by s i : X n − → X n .As defined above, L RX ( A ; C ) is a simplicial commutative augmented C -algebra spectrum.If M is an A -module spectrum, then L RX ( A ; M ) is defined. By slight abuse of notation wewon’t distinguish L RX ( A ; C ) or L RX ( A ; M ) from their geometric realization.If X is an arbitrary pointed simplicial set, then we can write it as the colimit of its fi-nite pointed subcomplexes and the Loday construction with respect to X can then also beexpressed as the colimit of the Loday construction for the finite subcomplexes.An important case is X = S n . In this case we write THH [ n ] ,R ( A ; C ) for L RS n ( A ; C ); this isthe higher order topological Hochschild homology of order n of A over R with coefficients in C .Let k be a commutative ring, A be a commutative k -algebra, and M be an A -module. Thenwe define THH [ n ] ,k ( A ; M ) := L HkS n ( HA ; HM ) . If A is flat over k , then π ∗ THH k ( A ; M ) ∼ = HH k ∗ ( HA ; HM ) [17, Theorem IX.1.7] and this alsoholds for higher order Hochschild homology in the sense of Pirashvili [35]: π ∗ THH [ n ] ,k ( A ; M ) ∼ = HH [ n ] ,k ∗ ( A ; M ) if A is k -flat [6, Proposition 7.2].To avoid visual clutter, given a commutative ring A and an element a ∈ A , we write A/a instead of A/ ( a ).2. Higher
THH of truncated polynomial algebras
When the Loday construction is viewed as a functor on pointed simplicial sets, it transformshomotopy pushouts of pointed simplicial sets into homotopy pushouts of Loday constructions.In [41, Section 3] Veen uses this to express higher
THH as a “topological Tor” of a lower
THH ,that is: for any commutative S -algebra A , THH [ n ] ( A ) ≃ A ∧ L THH [ n − ( A ) A. This yields a spectral sequence E s, ∗ = Tor THH [ n − ∗ ( A ) s ( A ∗ , A ∗ ) ⇒ THH [ n ] ∗ ( A ) . PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH In particular cases, this spectral sequence collapses for all n > THH [ n ] ∗ ( A ) as iterated Tor’s of A ∗ . In [6, Figures 1 and 2] we had a flow chart showing theresults of iterated Tor of F p over some F p -algebras with a particularly convenient form. Wecan do similar calculations over any field: Proposition 2.1. If F is a field of characteristic p and | ω | is even, there is a flow chart as inFigure 1 showing the calculation of iterated Tor ’s of F : If A is a term in the n th generation inthe flow chart, then Tor A ( F, F ) is the tensor product of all the terms in the ( n +1) st generationthat arrows from A point to. Here | ρ y | = | y | + 1 , | ǫz | = | z | + 1 and | ϕ z | = 2 + p | z | . F [ ω ] Λ( ǫω ) Γ( ρ ǫω ) ∼ = N k > F [ ρ k ǫω ] / ( ρ k ǫω ) p N k > Λ( ǫρ k ǫω ) N k > Γ( ϕ ρ k ǫω ) ∼ = N k,i > F [ ϕ i ρ k ǫω ] / ( ϕ i ρ k ǫω ) p · · ·· · ·· · · Figure 1.
Iterated Tor flow chart If F a field of characteristic zero, the analogous flow chart for | x | even is F [ x ] Λ( ǫx ) F [ ρ ǫx ] Λ( ǫρ ǫx ) · · · Proof.
In characteristic p , all divided power algebras split as tensor products of truncatedpolynomial algebras. This allows us to use the resolutions of [6, Section 2], where the tensorproducts in the respective bar constructions are all taken to be over F , to pass from eachstage to the next.In characteristic 0, the Tor dual of an exterior algebra is a divided power algebra, but thisis isomorphic to a polynomial algebra. Thus the resolutions of [6, Section 2], with the tensorproducts in the bar constructions again taken to be over F , can be used analogously to getthe alternation between exterior and polynomial algebras. (cid:3) Let x be a generator of even non-negative degree. In [6, Theorem 8.8] we calculated higher HH of truncated polynomial rings of the form F p [ x ] /x p ℓ for any prime p . The decompositiondue to B¨okstedt which is described before the statement of the theorem there does not workfor F p [ x ] /x m when m is not a power of p , but we can nevertheless use a similar kind ofargument to determine higher HH of F p [ x ] /x m as long as p divides m . This generalization of[6, Theorem 8.8] is interesting because if m is not a power of p , F p [ x ] /x m is no longer a Hopfalgebra, which the cases we discussed in [6] were.In the following HH [ n ] ∗ will denote Hochschild homology groups of order n whereas HH [ n ] denotes the corresponding simplicial object whose homotopy groups are HH [ n ] ∗ . BOBKOVA, H ¨ONING, LINDENSTRAUSS, POIRIER, RICHTER, AND ZAKHAREVICH
Theorem 2.2.
Let x be of even degree and let m be a positive integer divisible by p . Thenfor all n > HH [ n ] , F p ∗ ( F p [ x ] /x m ) ∼ = F p [ x ] /x m ⊗ B ′′ n ( F p [ x ] /x m ) , where B ′′ ( F p [ x ] /x m ) ∼ = Λ F p ( εx ) ⊗ Γ F p ( ϕ x ) , with | εx | = | x | + 1 , | ϕ x | = 2 + m | x | and B ′′ n ( F p [ x ] /x m ) ∼ = Tor B ′′ n − ( F p [ x ] /x m ) ∗ , ∗ ( F p , F p ) . Since F p [ x ] /x m is monoidal over F p , this gives a higher THH calculation,
THH [ n ] ∗ ( F p [ x ] /x m ) ∼ = THH [ n ] ∗ ( F p ) ⊗ HH [ n ] , F p ∗ ( F p [ x ] /x m ) ∼ = B n F p ( µ ) ⊗ F p [ x ] /x m ⊗ B ′′ n ( F p [ x ] /x m )(2.3)where B F p ( µ ) ∼ = F p [ µ ] with | µ | = 2 and B n F p ( µ ) = Tor B n − F p ( µ ) ∗ , ∗ ( F p , F p ) for n > Proof.
We use the standard resolution [24, (1.6.1)] and get that the Hochschild homology of F p [ x ] /x m is the homology of the complex . . . / / Σ m | x | F p [ x ] /x m ∆( x,x ) / / Σ | x | F p [ x ] /x m / / F p [ x ] /x m . Since p divides m , we have ∆( x, x ) = mx m − ≡ HH F p ∗ ( F p [ x ] /x m ) ∼ = F p [ x ] /x m ⊗ Λ F p ( εx ) ⊗ Γ F p ( ϕ x )at least as an F p [ x ] /x m -module, with | εx | = | x | + 1, | ϕ x | = 2 + m | x | . The map G. from [24,equation (1.8.6)] embeds this small complex quasi-isomorphically with its stated multiplicativestructure into the standard Hochschild complex for F p [ x ] /x m . It sends F p [ x ] /x m to itself indegree zero of the Hochschild complex and εx ⊗ x − x ⊗ , ϕ x m X i =1 1 X j =0 ( − j x i − j ⊗ x m − i ⊗ x j which generate an exterior and divided power subalgebra, respectively, inside the standardHochschild complex equipped with the shuffle product. The map G. is shown in [24] to behalf of a chain homotopy equivalence between the small complex and the standard Hochschildcomplex. So we get that HH F p ∗ ( F p [ x ] /x m ) = HH [1] , F p ∗ ( F p [ x ] /x m ) has the desired form as analgebra and sits as a deformation retract inside the standard complex calculating it.For the higher HH ∗ -computation we use that the E -term of the spectral sequence for HH [2] ∗ is E ∗ , ∗ = Tor F p [ x ] /x m ⊗ Λ F p ( εx ) ⊗ Γ F p ( ϕ x ) ( F p [ x ] /x m , F p [ x ] /x m )and as the generators εx and ϕ x come from homological degree one and two, the mod-ule structure of F p [ x ] /x m over Λ F p ( εx ) and Γ F p ( ϕ x ) factors over the augmentation to F p .Therefore the above Tor-term splits as(2.4) F p [ x ] /x m ⊗ Tor Λ F p ( εx ) ∗ , ∗ ( F p , F p ) ⊗ Tor Γ F p ( ϕ x ) ∗ , ∗ ( F p , F p ) . PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Now we can argue as in [6] to show that there cannot be any differentials or extensions inthis spectral sequence: although we are calculating the homology of the total complex of thebisimplicial F p -vector space of the bar construction B ( F p [ x ] /x m , HH F p ( F p [ x ] /x m ) , F p [ x ] /x m )which involves both vertical and horizontal boundary maps, we can map the bar construction B ( F p [ x ] /x m , F p [ x ] /x m ⊗ Λ F p ( εx ) ⊗ Γ F p ( ϕ x ) , F p [ x ] /x m )quasi-isomorphically into it, and the latter complex involves only non-trivial horizontal maps(all vertical differentials vanish) and has homology exactly equal to the algebra in equation(2.4). When all the vertical differentials in the original double complex are zero, there can beno nontrivial spectral sequence differentials d r for r >
2. Also, the trivial vertical differentialsmean that there can be no nontrivial extensions involving anything but the i th and ( i +1)st filtration, but since we can produce explicit generators whose p th powers (in the evendimensional case) or squares (in the odd dimensional case) actually vanish, we do not needto worry about extensions at all. Thus we obtain the claim about HH [2] , F p ∗ ( F p [ x ] /x m ).An iteration of this argument yields the result for higher Hochschild homology, since now weonly have exterior algebras on odd-dimensional classes and truncated algebras, truncated atthe p ’th power, on even-dimensional ones where the powers that vanish do so for combinatorialreasons not relating to the power of x that was truncated at in the original algebra. Ateach stage, the tensor factor F p [ x ] /x m will split off the E -term for degree reasons. Whatremains will be the Tor of F p with itself over a differential graded algebra that can be chosenup to chain homotopy equivalence to be a graded algebra A with a zero differential whichis moreover guaranteed by the flow chart to have the property that B ( F p , A, F p ) is chainhomotopy equivalent to its homology embedded as a subcomplex with trivial differentialinside it.The splitting for higher THH follows from the splitting for higher HH by arguing as in [6, 6.1](following [20, Theorem 7.1]) for the abelian pointed monoid { , x, . . . , x m − , x m = 0 } . (cid:3) Reducing the coefficients via the augmentation simplifies things even further. Here, theresult does not depend on the p -valuation of m , because x augments to zero and thereforeHochschild homology of F p [ x ] /x m with coefficients in F p is the homology of the complex . . . / / Σ m | x | F p ∆( x,x )=0 / / Σ | x | F p / / F p . Thus we obtain the following result.
Proposition 2.5.
For all primes p and for all m > HH [ n ] , F p ∗ ( F p [ x ] /x m ; F p ) ∼ = B ′′ n ( F p [ x ] /x m ) where B ′′ ( F p [ x ] /x m ) ∼ = Λ F p ( εx ) ⊗ Γ F p ( ϕ x ) and B ′′ n ( F p [ x ] /x m ) = Tor B ′′ n − ( F p [ x ] /x m ) ∗ , ∗ ( F p , F p ) for n > . Therefore we obtain (2.6) THH [ n ] ∗ ( F p [ x ] /x m ; F p ) ∼ = THH [ n ] ∗ ( F p ) ⊗ B ′′ n ( F p [ x ] /x m ) . This is shown as in the proof of the previous theorem using the method of [6], embedding εx ⊗ x ⊗ , ϕ x ⊗ x m − ⊗ x ⊗ BOBKOVA, H ¨ONING, LINDENSTRAUSS, POIRIER, RICHTER, AND ZAKHAREVICH inside the bar complex B ( F p , F p [ x ] /x m , F p ), where they generate exterior and divided poweralgebras, respectively, regardless of the divisibility of m . Remark . Note that the calculation becomes drastically different if ( p, m ) = 1 and we lookat the full HH [ n ] , F p ∗ ( F p [ x ] /x m ) rather than reducing coefficients to get HH [ n ] , F p ∗ ( F p [ x ] /x m ; F p ).Then multiplication by m is an isomorphism on F p [ x ] /x m -modules and hence HH F p ∗ ( F p [ x ] /x m ) ∼ = F p [ x ] /x m , for ∗ = 0 , (Σ | x | ( km +1) F p [ x ] /x m ) /x m − , for ∗ = 2 k + 1 , Σ km | x | ker( · x m − ) , for ∗ = 2 k, k > . A juggling formula
In this section we generalize juggling formulas from [19, § § R that can be different from the sphere spectrum and we relate the Lodayconstruction on a suspension on a pointed simplicial set X to the Loday construction on X .In [19] we mainly considered the cases where X is a sphere. Lemma 3.1.
Let X be a pointed simplicial set. For a sequence of cofibrations of commutative S -algebras S −→ R −→ A −→ B −→ C there is an equivalence of augmented commutative C -algebras. L AX ( B ; C ) ≃ C ∧ L L RX ( A ; C ) L RX ( B ; C ) . Proof.
For the duration of this proof smash products are formed over R , not S , but we stilldenote them by ∧ in order to simplify notation. For finite X the unit map of C , η C : R → C ,induces a map of coequalizer diagrams C ∧ ^ X n \∗ A ∧ ^ X n \∗ B ν R (cid:15) (cid:15) ν L (cid:15) (cid:15) id ∧ η C ∧ η C / / C ∧ C ∧ ^ X n \∗ A ∧ C ∧ ^ X n \∗ B n R (cid:15) (cid:15) n L (cid:15) (cid:15) C ∧ ^ X n \∗ B id ∧ η C / / C ∧ C ∧ ^ X n \∗ B Here, ν L sends the copies of A to C first and then multiplies C ∧ (cid:16)V X n \∗ C (cid:17) to C whereas ν R sends the copies of A to B and then uses the multiplication in each coordinate separately.The map n L sends the copies of A to C and multiplies C ∧ (cid:16) C ∧ V X n \∗ C (cid:17) to C sitting atthe basepoint whereas n R sends the copies of A to B and then uses the multiplication in B in PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH every coordinate. This morphism of coequalizer diagrams induces an isomorphism η on thecorresponding coequalizers, i.e. , η : ( L AX ( B ; C )) n −→ ( C ∧ L RX ( A ; C ) L RX ( B ; C )) n . This is an isomorphism because the diagram C = (cid:15) (cid:15) = (cid:15) (cid:15) id ∧ η C ∧ η C / / C ∧ C ∧ C µ ∧ id (cid:15) (cid:15) id ∧ µ (cid:15) (cid:15) C id ∧ η C / / C ∧ C induces the identity map between the coequalizers. A colimit argument proves the claim forgeneral X . (cid:3) Theorem 3.2 (Juggling Formula) . Let X be a pointed simplicial set. Then for any sequenceof cofibrations of commutative S -algebras S → R → A → B → C we get an equivalence ofaugmented commutative C -algebras L R Σ X ( B ; C ) ≃ L R Σ X ( A ; C ) ∧ L L AX ( C ) L BX ( C ) . Proof.
Consider the diagram C L RX ( A ; C ) / / o o C L RX ( C ) O O (cid:15) (cid:15) L RX ( A ; C ) / / o o O O (cid:15) (cid:15) C O O (cid:15) (cid:15) L RX ( C ) L RX ( B ; C ) / / o o C. By Lemma 3.1, taking the homotopy pushouts of the rows produces the diagram L R Σ X ( A ; C ) L AX ( C ) O O (cid:15) (cid:15) L BX ( C )whose homotopy pushout is L R Σ X ( A ; C ) ∧ L L AX ( C ) L BX ( C ) . We get an equivalent result by first taking the homotopy pushouts of the columns and thenof the rows. Homotopy pushouts on the columns produces C ∧ L L RX ( C ) L RX ( C ) L RX ( A ; C ) ∧ L L RX ( A ; C ) L RX ( B ; C ) o o / / C ∧ LC C which simplifies to C L RX ( B ; C ) o o / / C whose homotopy pushout is equivalent to L R Σ X ( B ; C ). (cid:3) Restricting our attention to spheres we obtain the following result. This is a relative variantof [19, Theorem 3.6].
Corollary 3.3.
Let S → R → A → B → C be a sequence of cofibrations of commutative S -algebras. Then for all n > there is an equivalence of augmented commutative C -algebras: THH [ n +1] ,R ( B ; C ) ≃ THH [ n +1] ,R ( A ; C ) ∧ L THH [ n ] ,A ( C ) THH [ n ] ,B ( C ) . Remark . The previous corollary gives a splitting of the same form as [19, Theorem 3.6].However, as the proof is different it is not obvious that the maps in the smash product arethe same. Thus (unlikely as it may be) it may turn out to be the case that this gives twodifferent but similar-looking splittings.3.1.
Beware the phony right-module structure!
In some cases it is tempting to usethe (valid) splitting of
THH [ n +1] ,R ( A ; C ) as THH [ n +1] ,R ( A ) ∧ A C and oversimplify the jug-gling formula we got in Corollary 3.3 to the invalid identification of THH [ n +1] ,R ( B ; C ) with THH [ n +1] ,R ( A ) ∧ A ( C ∧ L THH [ n ] ,A ( C ) THH [ n ] ,B ( C )) which in the case B = C becomes(3.5) THH [ n +1] ,R ( A ) ∧ A THH [ n +1] ,A ( C ) . This transformation is incorrect because it disregards the module structures, without whichthe maps of pushouts are not well-defined. The spectrum
THH [ n +1] ,R ( A ; C ) is not equivalentto THH [ n +1] ,R ( A ) ∧ A C as a right-module spectrum over THH [ n ] ,A ( C ). Assuming that therearrangement that leads to (3.5) were valid, any cofibration of commutative S -algebras S → A → B would produce an equivalence between THH [ n ] ( B ) and THH [ n ] ( A ) ∧ LA THH [ n ] ,A ( B ).But this equivalence does not hold in many examples, e.g. , for A = H Z and B equal to theEilenberg Mac Lane spectrum of F p or of the ring of integers in a number field.4. A generalization of Brun’s spectral sequence
In [9] Morten Brun uses the geometry of the circle to identify
THH ( HQ ; HQ ∧ LHk HQ )with THH ( Hk ; HQ ) where k is a commutative ring and Q is a commutative k -algebra: THH ( Hk ; HQ ) is a circle with HQ at the basepoint and Hk sitting at every non-basepointof S . Homotopy invariance says that we can let the point take over half the circle, so thatit covers an interval. This idea identifies THH ( Hk ; HQ ) with THH ( Hk ; B ( HQ, HQ, HQ ))where B denotes the two-sided bar construction. Brun then shows in [9, Lemma 6.2.3] thatthe latter is equivalent to THH ( HQ ; B ( HQ, Hk, HQ )) by a shift of perspective. This ideainspired our juggling formula 3.2 and also the following result.
Theorem 4.1 (Brun Juggling) . Let X be a pointed simplicial set. For any sequence of cofi-brations of commutative S -algebras S → R → A → B we get an equivalence of commutative PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH B -algebras L R Σ X ( A ; B ) ≃ B ∧ L L RX ( B ) L AX ( B ) . Note that B , which only appears at the basepoint on the left, now appears almost every-where on the right. Thus we can think of the basepoint as having “eaten” most of Σ X .In the following we will use the notation from [19, § Y is a pointed simplicial subset of X , then we denote by L R ( X,Y ) ( A, B ; B ) the relative Loday construction where we attach B toevery point in Y including the basepoint, A to every point in the complement and we use thestructure maps to turn this into a augmented commutative B -algebra spectrum. Note that if Y = ∗ , then L R ( X, ∗ ) ( A, B ; B ) = L RX ( A ; B ), so in this case we omit the ∗ from the notation asin Definition 1.1. Proof.
We consider the pair (Σ X, ∗ ) as ( CX ∪ X CX, CX ), with the cone sitting as the upperhalf of the suspension. Then, since the Loday construction is homotopy invariant, L R Σ X ( A ; B ) = L R Σ X, ∗ ( A, B ; B ) = L RCX ∪ X CX, ∗ ( A, B ; B ) ≃ L RCX ∪ X CX,CX ( A, B ; B )= L RCX ∪ X CX,CX ∪ X X ( A, B ; B ) . By [19, Proposition 2.10(b)] L RCX ∪ X CX,CX ∪ X X ( A, B ; B ) ≃ L RCX,CX ( A, B ; B ) ∧ L RX,X ( A,B ; B ) L RCX,X ( A, B ; B ) . By definition L RCX,CX ( A, B ; B ) = L RCX ( B ) and L RX ( B ) = L RX,X ( A, B ; B ) and by homotopyinvariance L RCX ( B ) ≃ B , hence L RCX ∪ X CX,CX ∪ X X ( A, B ; B ) ≃ B ∧ L RX ( B ) L RCX,X ( A, B ; B ) . Using [19, (3.0.1)] we can identify L RCX,X ( A, B ; B ) with(4.2) L RCX ( A ; B ) ∧ L L RX ( A ; B ) L RX ( B ; B )and as CX is contractible we obtain B ≃ L RCX ( A ; B ) and then Lemma 3.1 yields an equiva-lence of (4.2) with L AX ( B ). (cid:3) Example . Consider the case when X = S .(4.4) THH ( A ; B ) ≃ B ∧ B ∧ B ( B ∧ A B ) = THH ( B ; B ∧ A B ) . There is an Atiyah–Hirzebruch spectral sequence [17, IV.3.7] E p,q = π p ( E ∧ R Hπ q M ) = ⇒ π p + q ( E ∧ R M ) . Let B be a connective A -algebra. Setting R = B ∧ B , E = B and M = B ∧ A B we get E p,q = π p ( B ∧ B ∧ B Hπ q ( B ∧ A B )) = ⇒ π p + q ( B ∧ B ∧ B ( B ∧ A B )) . Setting B = HQ and A = Hk gives us E p,q = THH p ( Q ; Tor kq ( Q, Q )) = ⇒ π p + q ( HQ ∧ HQ ∧ HQ ( HQ ∧ Hk HQ )) ∼ = THH p + q ( k ; Q ) by (4.4). This recovers a spectral sequence with the same E page and limit as Brun’s [9,Theorem 6.2.10]. A substantial generalization of Brun’s spectral sequence for THH can befound in [22, Theorem 1.1].
Example . We can generalize Example 4.3 to any X . In particular, consider a commutativering k and a commutative k -algebra Q . If we apply the Atiyah–Hirzebruch spectral sequencein the case E = HQ R = L S n ( HQ ) M = L HkS n ( HQ )then the Brun juggling formula 4.1 gives us a spectral squence E p,q = π p ( HQ ∧ THH [ n ] ( Q ) H THH [ n ] ,kq ( Q )) = ⇒ THH [ n +1] p + q ( k ; Q ) . In the next section, we will see that we can identify
THH [ n ] ,k ( Q ) with higher order Shuklahomology, Sh [ n ] ,k ( Q ), so we get the simpler description E p,q = π p ( HQ ∧ THH [ n ] ( Q ) H ( Sh [ n ] ,kq ( Q )) = ⇒ THH [ n +1] p + q ( k ; Q ) . Corollary 4.6.
Let B be an augmented commutative A -algebra spectrum. Then applyingTheorem 3.2 to the sequence S = / / S / / A / / B / / A gives L Σ X ( B ; A ) ≃ L Σ X ( A ; A ) ∧ LA L BX ( A ) . In particular, if k is a commutative ring, A = Hk , and B = HQ for an augmented commu-tative k -algebra Q , then L Σ X ( HQ ; Hk ) ≃ L Σ X ( Hk ; Hk ) ∧ LHk L HQX ( Hk ) and if k is a field, then we obtain on the level of homotopy groups π ∗ L Σ X ( HQ ; Hk ) ∼ = π ∗ (Σ X ⊗ Hk ) ⊗ k π ∗ ( L HQX ( Hk )) . Remark . We stress that in Corollary 4.6 there is a spectrum level splitting of L Σ X ( HQ ; Hk )into L Σ X ( Hk ) smashed with an additional factor. In particular, for X = S n higher THH ofan augmented commutative k -algebra splits as THH [ n +1] ( Q ; k ) ≃ THH [ n +1] ( k ) ∧ LHk
THH [ n ] ,Q ( k ) . Greenlees proposed a splitting result in [18, Remark 7.2]: If k is a field and Q is a augmentedcommutative k -algebra, then his results yield a splitting THH ∗ ( Q ; k ) ≃ THH ∗ ( k ) ⊗ k Tor Q ∗ ( k, k ) . Our result generalizes his because for X = S the term L HQS ( Hk ) is nothing but Hk ∧ LHQ Hk whose homotopy groups are isomorphic to Tor Q ∗ ( k, k ). We will revisit this splitting result laterin Theorem 5.10, relating it to higher order Hochschild homology. PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Higher Shukla homology
Let k be a commutative ring. Ordinary Shukla homology [39] of a k -algebra A with coeffi-cients in an A -bimodule M can be identified with THH k ( A ; M ). We will define higher orderShukla homology in the context of commutative algebras as an iterated bar construction andidentify it with THH [ n ] ,k ( A ; M ) in Proposition 5.4. Definition 5.1.
Let A be a commutative k -algebra and B be a commutative A -algebra. Wedefine Sh [0] ,k ( A ; B ) = HA ∧ LHk
HB.
For n > n th order Shukla homology of A over k with coefficients in B as Sh [ n ] ,k ( A ; B ) = B S ( HB, Sh [ n − ,k ( A ; B ) , HB ) . where the latter is the two sided bar construction with respect to HB over the sphere spec-trum.Thus for n = 1 we have Sh [1] ,k ( A ; B ) ≃ THH k ( A ; B ). For example, when k = Z , p is aprime and a = p m , Sh Z ∗ ( Z /p m ; Z /p ) ∼ = Γ Z /p ( x ( m ))with | x ( m ) | = 2.It is consistent to set Sh [ − ,k ( A ) = Hk .A priori, THH k ( A ; B ) is a simplicial spectrum and Sh [ n ] ,k ( A ; B ) is therefore an n -simplicialspectrum, but we take iterated diagonals to get a simplicial spectrum and can then usegeometric realization to get an honest spectrum. Proposition 5.2.
Let R be a commutative ring and let a, p ∈ R be elements which are notzero divisors such that ( p ) is maximal and a ∈ ( p ) . Then Sh [0] ,R ∗ ( R/p ) ∼ = Λ R/p ( τ ) ∼ = Sh [0] ,R ∗ ( R/a ; R/p ) , | τ | = 1 and for n > , Sh [ n ] ,R ∗ ( R/p ) ∼ = Tor Sh [ n − ,R ∗ ( R/p ) ∗ ( R/p ; R/p ) and Sh [ n ] ,R ∗ ( R/a ; R/p ) ∼ = Tor Sh [ n − ,R ∗ ( R/a,R/p ) ∗ ( R/p ; R/p ) . Warning: the reduction
R/a → R/p does not induce an isomorphism Sh [ n ] ,R ∗ ( R/a ; R/p ) → Sh [ n ] ,R ∗ ( R/p ). By considering resolutions we can see that at n = 0 the induced map is themap taking τ to 0. In fact, in Corollary 7.4 we show that the map induced by R/a → R/p is zero on all generators other than the
R/p in dimension 0.
Proof.
We prove this by induction on n . At n = 0, Sh [0] ,R ( R/p ; R/p ) =
R/p ∧ LR R/p.
There is a K¨unneth spectral sequence, E s,t = Tor Rs,t ( R/p, R/p ) = ⇒ π s + t ( R/p ∧ LR R/p ) . We have a short resolution R · p −→ R → R/p, so Tor
Rs,t ( R/p, R/p ) ∼ = H s ( R/p −→ R/p ) t ∼ = R/p, s = 0 = t,R/p, s = 1 , t = 0 , , otherwise.For degree reasons, there cannot be any differentials or extensions in this spectral sequence,and the product of τ with itself has to vanish. Thus Sh [0] ∗ ( R/p ) ∼ = Λ R/p ( τ ), as desired. Notethat this proof works (almost) verbatim for Sh [0] ,R ∗ ( R/a ; R/p ).By [13, Proposition 2.1], as a augmented commutative
HR/p -algebra, Sh [0] ,R ( R/p ) ≃ HR/p ∨ Σ HR/p ≃ Sh [0] ,R ( R/a ; R/p ) . Thus Sh [1] ,R ( R/p ) = B ( HR/p, Sh [0] ,R ( R/p ) , HR/p ) ≃ HB ( R/p, Λ R/p ( τ ) , R/p ) . In the following let F be R/p . By [6], if we start with B F ( F , Λ F ( τ ) , F ) for F a field of positivecharacteristic, we know that the spectral sequence Tor Λ F ( τ ) ∗ , ∗ ( F , F ) ⇒ H ∗ ( B F ( F , Λ F ( τ ) , F ))collapses at E , which concludes the proof of the n = 1 case. We have that B F ( F , Λ F ( τ ) , F ) ≃ B ( F , Λ F ( τ ) , F ) because both calculate the homology of F ⊗ L Λ F ( τ ) F . Moreover, in [6] we showthat if we keep applying B F ( F , − , F ) to the result, having started with Λ F ( τ ), the spectralsequences Tor H ∗ ( − ) ∗ , ∗ ( F , F ) ⇒ H ∗ ( B F ( F , − , F )) will keep collapsing. Since in the case of acharacteristic zero field, a divided power algebra is isomorphic to a polynomial one, we canuse the method of [6], adjusted as in the proof of Proposition 2.1, to get the same result.Finally, in [13] we show that once we can exhibit a commutative H F -algebra as the imageof the Eilenberg Mac Lane functor on some simplicial algebra, we can continue doing thatwhen we apply B ( F , − , F ) to that algebra—once we get to the algebraic setting we can staythere. This concludes the proof for the collapsing of the spectral sequences both for R/p andfor
R/a . (cid:3) Definition 5.3.
For any commutative k -algebra A and any commutative A -algebra B wedefine higher derived Hochschild homology of A over k with coefficients in B , f HH [ n ] ,k ( A ; B ),as f HH [ n ] ,k ( A ; B ) = THH [ n ] ,k ( A ; B ) . Note that Sh [1] ,k ( A ; B ) = f HH [1] ,k ( A ; B ) = Sh k ( A ; B ). PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Proposition 5.4.
There is an isomorphism Sh [ n ] ,k ∗ ( A ; B ) ∼ = f HH [ n ] ,k ∗ ( A ; B ) for all n > . The idea for the following proof is due to Bjørn Dundas.
Proof.
By definition, the claim is true for n = 0. For higher n we have to show that the two-sided bar construction B S ( HB,
THH [ n ] ,k ( A ; B ) , HB ) is a model for THH [ n +1] ,k ( A ; B ); then theclaim follows by induction. Using the decomposition of the ( n + 1)-sphere as two hemispheresglued along the equator S n gives a homotopy pushout diagram for THH [ n +1] ,k ( A ; B ) THH [ n ] ,k ( A ; B ) (cid:15) (cid:15) / / HB (cid:15) (cid:15) HB / / THH [ n +1] ,k ( A ; B )in the model category of commutative Hk -algebras. The latter category is equivalent to thecategory of commutative S -algebras under Hk . The two-sided bar construction B S ( HB,
THH [ n ] ,k ( A ; B ) , HB )models the homotopy pushout HB ∧ L THH [ n ] ,k ( A ; B ) HB in the category of S -modules but this is also the homotopy pushout in the category of com-mutative S -algebras and in the category of commutative S -algebras under Hk ; here one canuse the model structure from [17] where cofibrant commutative S -algebras give the correcthomotopy type when involved in a smash product as an underlying S -module. (cid:3) Proposition 5.5.
In the special case of a sequence of cofibrations of commutative S -algebras R = Hk → Hk → HA → Hk with a cofibrant model of Hk and an augmented commutative k -algebra A we obtain (5.6) L Hk Σ X ( HA ; Hk ) ≃ L HAX ( Hk ) for any X .Proof. The juggling formula 3.2 for the sequence Hk → Hk → HA → Hk gives L Hk Σ X ( HA ; Hk ) ≃ L Hk Σ X ( Hk ; Hk ) ∧ L L HkX ( Hk ) L HAX ( Hk )but L HkX ( Hk ) ≃ Hk ≃ L Hk Σ X ( Hk ; Hk ). (cid:3) Remark . Note that Proposition 5.5 implies that L HAX ( Hk ) depends only on the homotopytype of Σ X , so L HAX ( Hk ) is a stable invariant of X . Consider the case where A is flat over k and X = S n . Then Equation (5.6) gives(5.8) L HkS n +1 ( HA ; Hk ) ≃ L HAS n ( Hk ) . The term on the left hand side of (5.8) has as homotopy groups the Hochschild homology oforder n + 1 of A with coefficients in k . The right hand side simplifies to THH [ n ] ,A ( k ) and thisis Shukla homology of order n of k over A . Therefore we obtain: Proposition 5.9.
Let k be a commutative ring and let A be an augmented commutative k -algebra which is flat as a k -module. Then for all n > HH [ n +1] ,k ∗ ( A ; k ) ∼ = Sh [ n ] ,A ∗ ( k ) . Note that for n = 0 we obtain the classical formula [10, X.2.1] HH k ∗ ( A ; k ) ∼ = Tor A ∗ ( k, k )Combining Proposition 5.9 with Corollary 4.6 we get the following splitting result for aug-mented commutative k -algebras. Theorem 5.10.
Let k be a commutative ring and let A be an augmented commutative k -algebra which is flat as a k -module. Then for all n > THH [ n ] ( A ; k ) ≃ THH [ n ] ( k ) ∧ Hk THH [ n ] ,k ( A ; k ) . If k is a field then we obtain the following isomorphism on the level of homotopy groups THH [ n ] ∗ ( A ; k ) ∼ = THH [ n ] ∗ ( k ) ⊗ k HH [ n ] ,k ∗ ( A ; k ) . A weak splitting for
THH ( R/ ( a , . . . , a r ); R/ m )Using a Tor-calculation by Tate from the 50’s we obtain a splitting on the level of homotopygroups of THH ∗ ( R/ ( a , . . . , a r ); R/ m ) in good cases. This yields an easy way of calculating THH ∗ ( Z /p m ; Z /p ) for m >
2. Compare [34, 9, 1] for other approaches.
Theorem 6.1.
Let R be a regular local ring with maximal ideal m and let ( a , . . . , a r ) be aregular sequence in R with a i ∈ m for i r . Then THH ∗ ( R/ ( a , . . . , a r ); R/ m ) ∼ = THH ∗ ( R ; R/ m ) ⊗ R/ m Γ R/ m ( S , . . . , S r ) with | S i | = 2 .Proof. Let I = ( a , . . . , a r ). Applying the juggling formula 3.2 to X = S and to the sequence S → HR → HR/I → HR/ m gives THH ( R/I ; R/ m ) ≃ THH ( R ; R/ m ) ∧ LHR/ m ∧ LHR
HR/ m ( HR/ m ∧ LHR/I
HR/ m ) . In [40] Tate determines the algebra structure on the homotopy groups of the last term,Tor
R/I ∗ ( R/ m , R/ m ) ∼ = Λ R/ m ( T , . . . , T d ) ⊗ R/ m Γ R/ m ( S , . . . , S r ) . Here, d is the dimension of m / m as an R/ m -vector space. We can choose a regular systemof generators ( t , . . . , t d ) for m such that the module structure of Tor R/I ∗ ( R/ m , R/ m ) over PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Tor R ∗ ( R/ m , R/ m ) ∼ = Λ R/ m ( T , . . . , T d ) is the canonical one (see [40, p. 22]). Hence the K¨unnethspectral sequence for THH ( R/I ; R/ m ) has an E -term isomorphic to THH ∗ ( R ; R/ m ) ⊗ R/ m Γ R/ m ( S , . . . , S r )which is concentrated in the zeroth column that consists of THH ∗ ( R ; R/ m ) ⊗ Tor R ∗ ( R/ m ,R/ m ) Tor
R/I ∗ ( R/ m , R/ m ) . Therefore, there are no non-trivial differentials and extensions in this spectral sequence. (cid:3)
We call the splitting of Theorem 6.1 a weak splitting because it is only a splitting on thelevel of homotopy groups. In Section 7 we develop a stronger spectrum-level splitting of asimilar form.We apply the above result in the special case where R is a principal ideal domain. Let p = 0 be an element of R , such that ( p ) is a maximal ideal in R and let n be bigger or equalto 2. Then we are in the situation of Theorem 6.1 because R ( p ) / ( p n ) ∼ = R/ ( p n ) so we candrop the assumption that R is local. The above result immediately gives an explicit formulafor THH ( R/ ( p n ); R/ ( p )). Corollary 6.2.
For all n > : THH ∗ ( R/ ( p n ); R/ ( p )) ∼ = THH ∗ ( R ; R/ ( p )) ⊗ R/ ( p ) Γ R/ ( p ) ( S ) . Remark . One may try to use the same method for
THH [ n ] . The juggling formula fromTheorem 3.2 for S = / / S / / H Z / / H Z /p m / / H Z /p gives us THH [ n ] ( Z /p m ; Z /p ) ≃ THH [ n ] ( Z ; Z /p ) ∧ L Sh [ n − , Z ( Z /p ) Sh [ n − , Z /p m ( Z /p ) . Thus we must understand the structure of Sh [ n − , Z /p m ( Z /p ) as a Sh [ n − , Z ( Z /p )-algebra. It isnot possible to do this through direct Tor computations for all n , as the computations rapidlybecome intractable; even Sh [1] , Z /p ( Z /p ) is rather involved [4, (5.2)], but see Proposition 7.5for a general formula.In order to obtain calculations in this example and in related cases, we need to develop themore delicate splitting of Section 7.7. A splitting for
THH [ n ] ( R/a ; R/p )Throughout this section, we assume that R is a commutative ring and a, p ∈ R are elementswhich are not zero divisors for which ( p ) is a maximal ideal and a ∈ ( p ) . Lemma 7.1.
Let R , p , and a be as above, and let π : R/a → R/p be the obvious reduction.Then the map induced by π , π ∗ : Sh [0] ,R ∗ ( R/a ; R/p ) −→ Sh [0] ,R ∗ ( R/p ) , factors as Sh [0] ,R ∗ ( R/a ; R/p ) ǫ / / R/p η / / Sh [0] ,R ∗ ( R/p ) . Proof.
The assumptions on a and p ensure that there exists a b ∈ R such that a = bp . Wehave the following diagram: R · a / / · bp (cid:15) (cid:15) R ǫ / / = (cid:15) (cid:15) R/aR · p / / R ǫ / / R/p
Thus we have a map of resolutions. When we tensor up with
R/p we get the following diagram: R ⊗ R R/p a ⊗ / / bp ⊗ (cid:15) (cid:15) R ⊗ R R/p = (cid:15) (cid:15) R ⊗ R R/p p ⊗ / / R ⊗ R R/p
We take the homology of the top and bottom row. Note that since a, p ∈ ( p ), the horizontalmaps are 0; thus the top and bottom row produce Tor’s which are of the form Λ R/p ( τ ).However, when we look at where τ goes from the top to the bottom, it maps by multiplicationby bp —which is 0 in R/p . Thus this map is 0. (cid:3)
Surprisingly enough, this special case allows us to prove a spectrum-level splitting for all n > Definition 7.2.
Let A HR/p be the category of augmented commutative
HR/p -algebras and h A HR/p its homotopy category. Let Mod
HR/p be the category of
HR/p -modules.
Lemma 7.3.
For R , p , and a as above, the map ϕ n : THH [ n ] ,R ( R/a ; R/p ) → THH [ n ] ,R ( R/p ) induced by R/a → R/p factors through
HR/p in h A HR/p .Proof.
The key step is the n = 0 case.From [13, Proposition 2.1] we know that THH [0] ,R ( R/a ; R/p ) ≃ HR/p ∨ Σ HR/p and also
THH [0] ,R ( R/p ) ≃ HR/p ∨ Σ HR/p . So we need to understand h A HR/p ( HR/p ∨ Σ HR/p, HR/p ∨ Σ HR/p ) . By [2, Proposition 3.2], we can identify this as h Mod
HR/p ( L Q R I ( HR/p ∨ Σ HR/p ) , Σ HR/p ) . Given an A ∈ A HR/p , we have a pullback IA (cid:15) (cid:15) / / A ǫ (cid:15) (cid:15) ∗ / / HR/p
PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Let B be a nonunital HR/p -algebra. Basterra defines Q ( B ) to be the pushout B ∧ HR/p B / / (cid:15) (cid:15) B (cid:15) (cid:15) ∗ / / Q ( B )We want to take the left- and right-derived versions of these functors for Basterra’s result.Let X be a fibrant replacement for HR/p ∨ Σ HR/p in A HR/p , so that X ǫ −→ HR/p is afibration. Thus the square R I ( HR/p ∨ Σ HR/p ) / / (cid:15) (cid:15) X ǫ (cid:15) (cid:15) (cid:15) (cid:15) ∗ / / HR/p is a homotopy pullback square (since every spectrum is fibrant [21, Proposition 13.1.2]). Forconciseness we write Y = R I ( HR/p ∨ Σ HR/p ). We have a long exact sequence of homotopygroups 0 → π Y → π X → π HR/p → π Y → π X → π HR/p → π − Y → , where we have used that X ≃ HR/p ∨ Σ HR/p so that its homotopy groups are concentratedin degrees 0 and 1. Note that the map π X → π HR/p is the identity. We thus see that π i Y ∼ = 0 for i = 1 and π Y ∼ = R/p .We need to identify h Mod
HR/p ( L Q (Σ HR/p ) , Σ HR/p ) ∼ = π F HR/p ( L Q (Σ HR/p ) , Σ HR/p ) , where F HR/p ( · , · ) is the function spectrum. We use the universal coefficient spectral sequence E s,t = Ext s,tR/p ( π ∗ L Q (Σ HR/p ) , π ∗ Σ HR/p )= ⇒ π t − s F HR/p ( L Q (Σ HR/p ) , Σ HR/p ) . Note that we’re working over a field, so E s,t = 0 for s = 0, and π ∗ (Σ HR/p ) is zero everywhereexcept at π , so in fact the spectral sequence collapses at E . Since we are only interested in π , the only term relevant to us is E , ∞ ∼ = E , ∼ = Hom R/p ( π L Q (Σ HR/p ) , R/p ) . Note that this group cannot be 0, since our hom-set contains at least two elements: theidentity map and the 0 map. Thus it remains to compute π L Q (Σ HR/p ).Consider the diagram (Σ
HR/p ) cof f / / ∼ (cid:15) (cid:15) (cid:15) (cid:15) L Q (Σ HR/p )Σ HR/p
By [3, Proposition 2.1], since (Σ
HR/p ) cof is 0-connected, f is 1-connected; thus π f is sur-jective. Since R/p is a field, we must have π L Q (Σ HR/p ) ∼ = 0 or R/p.
Since it can’t be 0, it must be
R/p , with the induced map being the identity.Let τ ( a )1 be the generator of the Λ( τ ) obtained as Sh [0] ,R ( R/a ; R/p ) and let τ ( p )1 be thegenerator of the Λ( τ ) obtained as Sh [0] ,R ( R/p ) in the calculation of Lemma 7.1. The abovecalculation shows that h Mod
HR/p ( L Q R I ( HR/p ∨ Σ HR/p ) , Σ HR/p ) ∼ = Hom R/p ( R/p, R/p )where the first copy of
R/p is generated by τ ( a )1 and the second copy is generated by τ ( p )1 . Butthe induced map on Sh [0] ,R takes τ ( a )1 to 0. Thus the corresponding map in h A R/p is also 0.This proves the n = 0 case.We now turn to the induction step. We have the composition B ( HR/p,
THH [ n − ,R ( R/a ; R/p ) , HR/p ) hyp . * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ B (1 ,ϕ n − , (cid:15) (cid:15) B ( HR/p, HR/p, HR/p ) t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ B ( HR/p,
THH [ n − ,R ( R/p ; R/p ) , HR/p )of maps of simplicial spectra. Taking the realization gives us the composition ϕ n : THH [ n ] ( R/a ; R/p ) −→ HR/p −→ THH [ n ] ( R/p ) , as desired. (cid:3) By applying π ∗ to the result of Lemma 7.3 we get the following generalization of Lemma 7.1. Corollary 7.4.
For R , p , and a as above, for all n > the map Sh [ n ] ,R ∗ ( R/a ; R/p ) −→ Sh [ n ] ,R ∗ ( R/p ) induced by R/a → R/p factors as Sh [ n ] ,R ∗ ( R/a ; R/p ) ǫ / / R/p η / / Sh [ n ] ,R ∗ ( R/p ) . Here,
R/p is considered as a graded ring concentrated in degree ; the first map in the fac-torization is the augmentation and the second is the unit map induced by the inclusion of thebasepoint. By Lemma 3.1,
THH [ n ] ,R/a ( R/p ) ≃ HR/p ∧ THH [ n ] ,R ( R/a ; R/p ) THH [ n ] ,R ( R/p ) . PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH However, by Lemma 7.3 the map
THH [ n ] ,R ( R/a ; R/p ) → THH [ n ] ,R ( R/p ) factors through
HR/p . This proves the following result about higher order Shukla homology:
Proposition 7.5.
For R , p , and a as above, THH [ n ] ,R/a ( R/p ) ≃ ( HR/p ∧ THH [ n ] ,R ( R/a ; R/p ) ∧ HR/p ) ∧ HR/p
THH [ n ] ,R ( R/p ) ≃ THH [ n +1] ,R ( R/a ; R/p ) ∧ R/p
THH [ n ] ,R ( R/p ) . This recovers the calculation of Sh Z /p ∗ ( Z /p ) from [4, 5.2]. It also explains why these Shuklacalculations are more involved than Shukla homology calculations of the form Sh R ∗ ( R/x ) where x is a regular element. In the latter case we just obtain a divided power algebra over R/x ona generator of degree two, whereas for all m > Sh Z /p m ∗ ( Z /p ) ∼ = Sh [2] , Z ∗ ( Z /p m ; Z /p ) ⊗ Z /p Sh Z ∗ ( Z /p ) ∼ = O i > Λ( ε ( ̺ i ( τ ( p m )1 ))) ⊗ Γ Z /p ( ϕ ̺ i ( τ ( p m )1 ))) ⊗ Z /p Γ Z /p ( ̺ ( τ ( p )1 )) . We are now ready to prove the main splitting result:
Theorem 7.6. If R is a commutative ring and if p, a ∈ R are elements which are not zerodivisors for which ( p ) is a maximal ideal and a ∈ ( p ) , then THH [ n ] ( R/a ; R/p ) ≃ THH [ n ] ( R ; R/p ) ∧ LHR/p
THH [ n ] ,R ( R/a ; R/p ) . Proof.
Recall that in the category of commutative algebras, the smash product is the sameas the pushout. Consider the following diagram:
HR/p / / (cid:15) (cid:15) THH [ n − ,R ( R/p ; R/p ) / / (cid:15) (cid:15) THH [ n ] ( R ; R/p ) (cid:15) (cid:15) THH [ n ] ,R ( R/a ; R/p ) / / THH [ n − ,R/a ( R/p ; R/p ) / / THH [ n ] ( R/a ; R/p ) . By Proposition 7.5 the left square is a homotopy pushout square and the right square is ahomotopy pushout square by the juggling formula 3.2, with the maps of those formulas. Thusthe outside of the diagram also gives a homotopy pushout, producing the formula
THH [ n ] ( R/a ; R/p ) ≃ THH [ n ] ( R ; R/p ) ∧ LHR/p
THH [ n ] ,R ( R/a ; R/p ) , as desired. (cid:3) The Loday construction of products
We establish a splitting formula for Loday constructions of products of ring spectra. Thisresult is probably well-known, but as we will need it later, we provide a proof. In the case of X = S such a splitting is proved for connective ring spectra in [15] in the context of ’ringfunctors’. See also [12, Proposition 4.2.4.4].For two ring spectra A and B we consider their product A × B with the multiplication( A × B ) ∧ ( A × B ) → A × B that is induced by the maps ( A × B ) ∧ ( A × B ) → A and ( A × B ) ∧ ( A × B ) → B that aregiven by the projection maps to A and B and the multiplication on A and B :( A × B ) ∧ ( A × B ) pr A ∧ pr A / / pr B ∧ pr B (cid:15) (cid:15) A ∧ A µ A / / AB ∧ B µ B (cid:15) (cid:15) B For X = S we obtain L S ( A × B ) = ( A × B ) ∧ ( A × B )and this is equivalent to A ∧ A ∨ A ∧ B ∨ B ∧ A ∨ B ∧ B whereas L S ( A ) × L S ( B ) is equivalentto A ∧ A ∨ B ∧ B so in this case L S ( A × B ) is not equivalent to L S ( A ) ∨ L S ( B ). In general,if a simplicial set has finitely many connected components, say X = X ⊔ . . . ⊔ X n , then L X ( A × B ) ≃ L X ( A × B ) ∧ . . . ∧ L X n ( A × B )so it suffices to study L X ( A × B ) for connected simplicial sets X . We will first considerthe case X = S , where L S with respect to the minimal simplicial model of the circle is THH = THH [1] , and then use that special case to prove the result for general connected finitesimplicial sets X .We thank Mike Mandell who suggested to use Brooke Shipley’s version of THH in thesetting of symmetric spectra. Brooke Shipley shows in [37] that a variant of B¨okstedt’s modelfor
THH in symmetric spectra of simplicial sets is equivalent to the version that mimics theHochschild complex and she proves several important features of this construction. See also[33] for a correction of the proof of the comparison.
Proposition 8.1.
For symmetric ring spectra A and B , the product of the projections inducesa stable equivalence THH ( A × B ) → THH ( A ) × THH ( B ) . In the following proof we denote by
THH the model of
THH defined in [37, Definition 4.2.6].This is no abuse of notation: Let A and B be S -cofibrant symmetric ring spectra [38, Theorem2.6]. Then by [37, Theorem 4.2.8] and [33, Theorem 3.6] THH ( A ) and THH ( B ) are stablyequivalent to THH ( A ) and THH ( B ) in our sense. As THH ( − ) sends stable equivalences tostable equivalences ([37, Corollary 4.2.9]) we can choose an S -cofibrant replacement of A × B ,( A × B ) c , and get that THH ( A × B ) is stably equivalent to THH (( A × B ) c ) and this in turnis stably equivalent to our notion of THH . Proof.
Note that for any symmetric ring spectrum R , THH ( R ) is defined as the diagonal of abisimplicial symmetric spectrum THH • ( R ) [37, 4.2.6], where one of the simplicial directionscomes from the THH -construction and the other one comes from the fact that we are workingwith symmetric spectra in simplicial sets. In [33, p. 4101] the authors use the geometricrealization instead of the diagonal, but this does not cause any difference in the arguments.
PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH We will start by showing that there is a chain of stable equivalences between
THH ( A × B )and THH ( A ) × THH ( B ). We use the following chain of identifications: THH ( A × B ) (pr A , pr B ) (cid:15) (cid:15) hocolim ∆ op THH • ( A × B ) ≃ o o hocolim ∆ opf THH • ( A × B ) ≃ o o hocolim ∆ opf THH • ( A ∨ B ) ≃ O O R (cid:15) (cid:15) hocolim ∆ opf ( THH • ( A ) ∨ THH • ( B )) ≃ (cid:15) (cid:15) J O O hocolim ∆ opf THH • ( A ) ∨ hocolim ∆ opf THH • ( B ) ≃ (cid:15) (cid:15) THH ( A ) × THH ( B ) THH ( A ) ∨ THH ( B ) ≃ o o hocolim ∆ op THH • ( A ) ∨ hocolim ∆ op THH • ( B ) ≃ o o
1) With [37, Proposition 4.27] we obtain a level equivalence between
THH ( A × B ) andhocolim ∆ op THH • ( A × B ). Similarly, in the bottom row we can identify the homotopy colimitwith THH . This does not need any cofibrancy assumptions.2) Let ∆ f denote the subcategory of ∆ containing all objects but only injective maps. Theinduced map on homotopy colimitshocolim ∆ opf THH • ( A × B ) → hocolim ∆ op THH • ( A × B )is an equivalence because the homotopy colimit of symmetric spectra is defined levelwise [37,Definition 2.2.1] and in every simplicial degree p and every level ℓ , THH p ( A × B )( ℓ ) is cofibrant(because it is just a simplicial set) and hence the claim follows as in [11, Proposition 20.5].An analoguous argument applies in the second occurence of 2).3) We consider A ∨ B as a non-unital commutative ring spectrum via the multiplicationmap( A ∨ B ) ∧ ( A ∨ B ) ≃ ( A ∧ A ) ∨ ( A ∧ B ) ∨ ( B ∧ A ) ∨ ( B ∧ B ) → ( A ∧ A ) ∨ ( B ∧ B ) → A ∨ B where the first map sends the mixed terms to the terminal spectrum and the second one usesthe multiplication in A and B . Correspondingly, THH • ( A ∨ B ) is a presimplicial spectrum,that only uses the face maps of the structure maps of THH .The canonical map A ∨ B → A × B is a stable equivalence of non-unital symmetric ringspectra and hence by adapting the argument in the proof of [37, Corollary 4.2.9] we get a π ∗ -equivalence of the corresponding presimplicial objects THH • ( A ∨ B ) ≃ THH • ( A × B ) . Pointwise level equivalences give level equivalences on homotopy colimits [37, Proposition2.2.2], so the map in 3) is a level equivalence.
4) Homotopy colimits commute with sums.5) The product is stably equivalent to the sum.To have a chain of stable equivalences between
THH ( A × B ) and THH ( A ) × THH ( B ) itremains to understand the effect of the maps J and R and we will control them in Lemma8.2 below.We claim that the product of the projectionspr A : A × B → A, pr B : A × B → B produces an equivalence. Observe that we can apply the projection pr A to every stage inour diagram. On A ∨ B , this will induce the collapse map A ∨ B → A . By applying pr A tothe entire chain of equivalences, we get a diagram of equivalences between various versions of THH ( A ). We can do the same for pr B . This gives a commutative diagram THH ( A × B ) (pr A , pr B ) (cid:15) (cid:15) chain of maps THH ( A ) × THH ( B ) pr A × pr B (cid:15) (cid:15) THH ( A ) × THH ( B ) (pr A (the chain) , pr B (the chain)) THH ( A ) × THH ( B )where the chain is a zigzag of arrows going both ways. The chain on top has all its stagesequivalences. By the above discussion, so is the product of the chains on the bottom. Andthe map on the right is the identity. So working step by step in the zigzag from the right,we show that the pair of projections (pr A , pr B ) induces equivalences at all the intermediatesteps, until we get to the leftmost (pr A , pr B ) which is therefore also an equivalence. (cid:3) We consider the map j : THH • ( A ) ∨ THH • ( B ) → THH • ( A ∨ B )that is induced by the inclusions A ֒ → A ∨ B and B ֒ → A ∨ B . We let J be the induced mapon the homotopy colimit. It has a retraction R = hocolim ∆ opf r with r : THH • ( A ∨ B ) → THH • ( A ) ∨ THH • ( B )that sends all mixed smash products to the terminal object. Note that R ◦ J = id. Lemma 8.2.
There is a presimplicial homotopy j ◦ r ≃ id .Proof. We consider the n th presimplicial degree of THH n ( A ∨ B ): THH n ( A ∨ B ) = ( A ∨ B ) n +1 . This is a sum of terms of the form A ∧ i ∧ B ∧ i ∧ . . . ∧ B ∧ i k ∧ A ∧ i k +1 for suitable k with0 i , i k +1 and 0 < i j for 1 < j < k + 1, so that P k +1 j = . i j = n + 1.Restricted to such a summand we define h j : ( A ∨ B ) n +1 → ( A ∨ B ) n +2 for 0 j n as h j | A ∧ i ∧ B ∧ i ∧ ... ∧ B ∧ ik ∧ A ∧ ik +1 = id A ∧ j +1 ∧ η A ∧ id , if j + 1 i , id B ∧ j +1 ∧ η B ∧ id , if i = 0 and j + 1 i , ∗ , otherwise. PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Then d h = id, d n +1 h n = j ◦ r and d i h j = h j − d i , i < j,d i h j − , i = j = 0 ,h j d i − , i > j + 1 . (cid:3) This implies that J ◦ R ≃ id, so we get thathocolim ∆ opf THH • ( A ∨ B ) ≃ hocolim ∆ opf ( THH • ( A ) ∨ THH • ( B )) . For the general setting we work with commutative ring spectra and we return to the settingof [17]. Note that the naturality of cofibrant replacements ensures that we get morphisms ofcommutative ring spectra A c ← ( A × B ) c → B c and hence a weak equivalence (because the product of acyclic fibrations is an acyclic fibration).( A × B ) c → A c × B c . This proves the case of X = ∗ of the following proposition and is needed in the proof. Proposition 8.3.
For any connected finite simplicial set X , the projection maps pr A : A × B and pr B : A × B → B induce an equivalence L X (( A × B ) c ) ≃ L X ( A c ) × L X ( B c ) and in particular, for all n > , THH [ n ] (( A × B ) c ) ≃ THH [ n ] ( A c ) × THH [ n ] ( B c ) . Proof.
We prove the result for all finite connected simplicial sets X by induction on thedimension n of the top non-degenerate simplex in X . Since the only finite connected simplicialset with its only non-degenerate simplices in dimension zero is a point, the result is obviousfor n = 0.For higher n , the crucial observation is that if we have simplicial sets X , Y , and Z so that Z is a non-empty subset of both X and Y , then if the projection maps ( A × B ) c → A c and( A × B ) c → B c induce equivalences as given in the statement of this proposition for X , Y and Z , then we also obtain an equivalence(8.4) L X ∪ Z Y (( A × B ) c ) ≃ / / L X ∪ Z Y ( A c ) × L X ∪ Z Y ( B c ) . This is because then L X ∪ Z Y (( A × B ) c ) ≃ L X (( A × B ) c ) ∧ L Z (( A × B ) c ) L Y (( A × B ) c ) ≃ ( L X ( A c ) × L X ( B c )) ∧ L L Z ( A c ) ×L Z ( B c ) ( L Y ( A c ) × L Y ( B c )) ≃ ( L X ( A c ) ∧ L Z ( A c ) L Y ( A c )) × ( L X ( B c ) ∧ L Z ( B c ) L Y ( B c )) ≃ L X ∪ Z Y ( A c ) × L X ∪ Z Y ( B c ) . For the first and last equivalence we use that the Loday construction sends pushouts tohomotopy pushouts and that for a cofibrant commutative ring spectrum the map L Z ( R ) →L Y ( R ) is a cofibration. For the second equivalence we use our assumption that the propositionholds for X , Y and Z .The third equivalence holds because L Z ( A c ) acts trivially on L X ( B c ) and on L Y ( B c ) thus itsends the corresponding factors to the terminal ring spectrum; the same holds for the action of L Z ( B c ) on L X ( A c ) and on L Y ( A c ). Therefore a K¨unneth spectral sequence argument showsthat we obtain a weak equivalence.For n = 1, we use homotopy invariance of the Loday construction and the fact that anyfinite connected simplicial set with non-degenerate simplices only in dimensions 0 and 1 ishomotopy equivalent to W mi =1 S for some m >
0. If m = 0, we deduce the proposition fromthe n = 0 case above; if m = 1, we use Proposition 8.1, and for m >
1, we use induction andEquation (8.4).For the inductive step, assume that n > < n , and in particularfor ∂ ∆ n , the boundary of the standard n -simplex. Assume that we have a finite simplicialset X for which the proposition holds. We then prove that the proposition also holds for X ∪ ∂ ∆ n ∆ n , that is: X with an additional n -simplex glued to it along the boundary. Withoutloss of generality, we may assume that the boundary of the new n -simplex is embedded in X : if it is not, apply four-fold edgewise subdivision to everything, and then X ∪ ∂ ∆ n ∆ n willconsist of the central small n -simplex inside the original n -simplex that was added that doesnot touch the boundary of the originally added n -simplex and all the rest of the subdividedcomplex. But the rest of the subdivided complex is homotopy equivalent to the original X ,so the proposition holds for it, and the central small n -simplex does indeed have its boundaryembedded in the four-fold edgewise subdivision of X ∪ ∂ ∆ n ∆ n .Then by assumption, the proposition holds for X , by the inductive hypothesis it holds for ∂ ∆ n , by homotopy invariance it holds for ∆ n ≃ ∗ , and so by Equation (8.4) it holds for X ∪ ∂ ∆ n ∆ n . (cid:3) For later use we need a version of Proposition 8.3 with coefficients. Again, we choosecofibrant models ( A × B ) c , A c and B c and we assume M c is a cofibrant A c -module spectrum, N c is a cofibrant B c -module spectrum and ( M × N ) c is a cofibrant ( A × B ) c -module spectrum,such that these cofibrant replacements are compatible with the projection maps on ( A × B ) c and ( M × N ) c . Corollary 8.5.
For all connected pointed finite simplicial sets there is an equivalence L X (( A × B ) c ; ( M × N ) c ) → L X ( A c ; M c ) × L X ( B c ; N c ) . Proof.
The argument in the proof of Proposition 8.3 can be adapted to pointed finite simplicialsets. We know that L X (( A × B ) c ; ( M × N ) c ) ≃ L X (( A × B ) c ) ∧ ( A × B ) c (( M × N ) c ) PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH and by the result above this is equivalent to( L X ( A c ) × L X ( B c )) ∧ LA c × B c ( M c × N c )Again, we can identify the coequalizers because the action of A c on N c and the one of B c on M c is trivial and obtain( L X ( A c ) ∧ A c M c ) × ( L X ( B c ) ∧ B c N c ) ≃ L X ( A c ; M c ) × L X ( B c ; N c ) . (cid:3) Applications
THH [ n ] ( Z /p m ; Z /p ) . This example was our original motivation for obtaining the splittingresult of Theorem 7.6. We apply it to the case where R = Z , p is a prime, and a = p m for m >
2. As a special case of Theorem 7.6 we obtain the following splitting.
Theorem 9.1.
THH [ n ] ( Z /p m ; Z /p ) ≃ THH [ n ] ( Z ; Z /p ) ∧ LH Z /p THH [ n ] , Z ( Z /p m ; Z /p ) ∼ = THH [ n ] ( Z ; Z /p ) ∧ LH Z /p Sh [ n ] , Z ( Z /p m ; Z /p ) . This gives a direct calculation of
THH [ n ] ∗ ( Z /p m ; Z /p ) for all n because in [13, Theorem 3.1]we determine THH [ n ] ∗ ( Z ; Z /p ) as an iterated Tor-algebra B n F p ( x ) ⊗ F p B n +1 F p ( y ) where | x | = 2 p , | y | = 2 p − B F p ( z ) = F p [ z ] and B n F p ( z ) = Tor B n − F p ( z ) ∗ ( F p , F p ). We determined Sh [ n ] , Z ( Z /p m ; Z /p )in Proposition 5.2.The case n = 1 was calculated in [34], [9] and later as well in [1]. Remark . Note that we cannot use the sequence canonical projection maps . . . / / Z /p m +1 Z / / Z /p m Z / / . . . / / Z /p Z / / Z /p Z in order to compare the groups THH ( H Z /p m Z ; H Z /p Z ) for varying m because the tensor fac-tors coming from THH ∗ ( H Z ; H Z /p Z ) are mapped isomorphically from THH ( H Z /p m +1 Z ; H Z /p Z )to THH ( H Z /p m Z ; H Z /p Z ) whereas the tensor factor Sh [ n ] , Z ( Z /p m +1 ; Z /p ) is mapped via theaugmentation map to Sh [ n ] , Z ( Z /p m ; Z /p ) in each step of the sequence. This is straightforwardto see with the help of the explicit resolutions used in the proof of Lemma 7.3.9.2. Number rings.
As a warm-up we consider R = Z [ i ], p = 1 − i and 2 ∈ ( p ) . Then weget that THH [ n ] ( Z [ i ] / Z [ i ] / (1 − i )) ≃ THH [ n ] ( Z [ i ]; Z [ i ] / (1 − i )) ∧ H Z [ i ] / (1 − i ) Sh [ n ] , Z [ i ] ( Z [ i ] / Z [ i ] / (1 − i )) . Note that Z [ i ] / (1 − i ) ∼ = Z / Z [ i ] / ∼ = F [ x ] /x . Thus we can calculate THH [ n ] ( Z [ i ] / Z [ i ](1 − i )) ∼ = THH [ n ] ( F [ x ] /x ; F )using the flow chart in [6] and we know from [13, Theorem 4.3] that THH [ n ] ∗ ( Z [ i ]; Z [ i ] / (1 − i ))can also be computed using iterated Tor’s. The term Sh [ n ] , Z [ i ] ( Z [ i ] / Z [ i ] / (1 − i )) can becomputed as an iterated Tor by Proposition 5.2. Thus all of the components of the above expression are known. What was not known before is that THH [ n ] ( Z [ i ] / Z [ i ] / (1 − i )) splitsin the above manner.The general case is as follows: Consider p ∈ Z a prime, and let K be a number field suchthat p is ramified in O K , with p = p e · · · p e r r with one e i >
1. The Chinese RemainderTheorem let’s us split O K /p as a ring as O K /p ∼ = r Y j =1 O K / p e j j and O K / p i as an O K /p -module is then isomorphic to 0 × . . . × O k / p i × . . . × i . With Corollary 8.5 we obtain the followingresult. Theorem 9.3. (9.4)
THH [ n ] ( O K /p ; O K / p i ) ≃ THH [ n ] ( O K ; O / p i ) ∧ H O K / p i THH [ n ] , O K ( O K /p ; O K / p i ) . Again, O K /p ∼ = ( O K ) p i /p is isomorphic to O K / p i [ π ] /π e i where π is the uniformizer,hence O K / p i [ π ] /π e i ∼ = O K / p i [ x ] /x e i so we can calculate THH [ n ] ( O K /p ; O K / p i ) using Propo-sition 2.5. We can determine THH [ n ] ( O K ; O / p i ) using [13, Theorem 4.3] and we calculated THH [ n ] , O K ( O K /p ; O K / p i ) in Proposition 5.2. Using these calculations one can deduce rightaway that there is a splitting on the level of homotopy groups of THH [ n ] ∗ ( O K /p ; O K / p i ).But (9.4) yields a splitting of THH [ n ] ( O K /p ; O K / p i ) on the level of augmented commutative H O K / p i -algebra spectra.9.3. Galois descent.
In [36, Definition 9.2.1] John Rognes defines a map of commutative S -algebras f : A → B to be formally THH -´etale, if the unit map B → THH A ( B ) is a weakequivalence. Note that this implies that the augmentation map THH A ( B ) → B that isinduced by multiplying all B -entries in THH A ( B ) together, is also a weak equivalence becausethe compositve B → THH A ( B ) → B is the identity on B .Therefore, applying the Brun juggling formula 4.1 in this case to X = S we obtain THH [2] ( A ) ∧ LA B ≃ THH [2] ( A ; B ) ≃ B ∧ THH ( B ) THH A ( B ) ≃ B ∧ L THH ( B ) B ≃ THH [2] ( B ) . We can slightly generalize this:
Definition 9.5.
Let X be a pointed simplicial set. A morphism f : A → B is formally X -´etale, if the unit map B → L AX ( B ) is a weak equivalence.For formally X -´etale morphisms f : A → B the Brun juggling formula 4.1 for X implies L Σ X ( A ) ∧ LA B ≃ L Σ X ( A ; B ) ≃ B ∧ L X ( B ) L AX ( B ) ≃ B ∧ L X ( B ) B ≃ L Σ X ( B ) . This statement is related to Akhil Mathew’s result [30, Proposition 5.2] where he showsthat L Y ( A ; B ) ≃ L Y ( B ) if f : A → B is a faithful finite G -Galois extension and if Y is asimply-connected pointed simplicial set. Such Galois extensions are formally THH -´etale by[36, Lemma 9.2.6].
PLITTINGS AND CALCULATIONAL TECHNIQUES FOR HIGHER
THH Algebraic function fields over F p . In several of our splitting formulas higher
THH ofthe ground field is a tensor factor. So far we have only considered prime fields or rather simple-minded algebraic extensions of those. Topological Hochschild homology groups of algebraicfunction fields are an important class of examples.Let L be an algebraic function field over F p . Then there is a transcendence basis ( x , . . . , x d )such that L is a finite separable extension of F p ( x , . . . , x d ) [32, Theorem 9.27]. As separableextensions do not contribute anything substantial to topological Hochschild homology weobtain the following result: Theorem 9.6.
Let L be an algebraic function field over F p , then THH ( L ) ∗ ∼ = L ⊗ F p THH ∗ ( F p ) ⊗ F p Λ F p ( εx , . . . , εx d ) . Proof.
McCarthy and Minasian show in [31, 5.5, 5.6] that
THH has ´etale descent in our case.Therefore
THH ( L ) ≃ HL ∧ LH F p ( x ,...,x d ) THH ( F p ( x , . . . , x d )) ≃ HL ∧ LH F p ( x ,...,x d ) H F p ( x , . . . , x d ) ∧ LH F p [ x ,...,x d ] THH ( F p [ x , . . . , x d ]) ≃ HL ∧ LH F p [ x ,...,x d ] THH ( F p [ x , . . . , x d ]) . But the topological Hochschild homology of monoid rings is known by [20, Theorem 7.1] andhence π ∗ THH ( F p [ x , . . . , x d ]) ∼ = THH ∗ ( F p ) ⊗ F p HH ∗ ( F p [ x , . . . , x d ]). As HH ∗ ( F p [ x , . . . , x d ]) ∼ = F p [ x , . . . , x d ] ⊗ F p Λ F p ( εx , . . . , εx d )we get the result. (cid:3) McCarthy and Minasian actually show more in [31, 5.5,5.6], and we can adapt the aboveproof to a more general situation.
Theorem 9.7.
Let X be a connected pointed simplicial set X . Then L X ( HL ) ≃ HL ∧ LH F p [ x ,...,x d ] L X ( F p [ x , . . . , x d ]) . The Loday construction on pointed monoid algebras satisfies a splitting of the form L X ( H F p [Π + ]) ≃ L X ( H F p ) ∧ H F p L H F p X ( H F p [Π + ]) , see [20, Theorem 7.1]. Therefore L X ( F p [ x , . . . , x d ]) splits as L X ( H F p ) ∧ H F p L H F p X ( H F p [ x , . . . , x d ]).In particular, for X = S n we get an explicit formula for THH [ n ] ( L ): Corollary 9.8.
For all n > : THH [ n ] ( L ) ≃ HL ∧ LH F p [ x ,...,x d ] ( THH [ n ] ( F p ) ∧ H F p THH [ n ] , F p ( F p [ x , . . . , x d ])) . Recall that we know π ∗ THH [ n ] , F p ( F p [ x , . . . , x d ]) = HH [ n ] , F p ∗ ( F p [ x , . . . , x d ]) ∼ = HH [ n ] , F p ∗ ( F p [ x ] ⊗ F p d ) ∼ = HH [ n ] , F p ∗ ( F p [ x ]) ⊗ F p d and we determined HH [ n ] , F p ∗ ( F p [ x ]) in [6, Theorem 8.6]. Remark . Topological Hochschild homology of L considers HL as an S -algebra and thisallows us to consider L over the prime field. The Hochschild homology of an algebraic functionfield L over a general field K was for instance determined in [8, Corollary 5.3] and is morecomplicated. Remark . Note that Theorem 9.6 contradicts the statement of [18, Remark 7.2]. In thatremark, it is crucial to assume that one works in an augmented setting; in the above situation,this is not the case.9.5.
THH [ n ] ( Z /p m ) . We close with the open problem of computing
THH [ n ] ( Z /p m ) for higher n . The juggling formula 3.2 applied to the sequence S → H Z → H Z /p m = H Z /p m for m > THH [ n ] ( Z /p m ) ≃ THH [ n ] ( Z ; Z /p m ) ∧ L THH [ n − , Z ( H Z /p m ) H Z /p m . Up to n = 2 we know THH [ n ] ∗ ( H Z ): The case n = 1 is B¨okstedt’s calculation [7] and n = 2is [14, Theorem 2.1]. Therefore we can determine THH [ n ] ∗ ( Z ; Z /p m ) up to n = 2. As p m isregular in Z , THH Z ∗ ( Z /p m ) = Sh Z ∗ ( Z /p m ) ∼ = Γ Z /p m ( x )with | x | = 2. If we could determine the right Sh Z ∗ ( Z /p m )-module structure on THH [2] ∗ ( Z ; Z /p m ),then this would allow us to calculate the E -term of the K¨unneth spectral sequence for THH [2] ∗ ( Z /p m ), E p,q = Tor Γ Z /pm ( x ) p,q ( THH [2] ∗ ( Z ; Z /p m ) , Z /p m ) ⇒ THH [2] ∗ ( Z /p m ) . References [1] Vigleik Angeltveit,
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School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
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