1 Topological Hochschild homology of S-algebras
1.3 First properties of topological Hochschild homology
An important example is the topological Hochschild homology of an S-algebra coming from a (simplicial) ring. We consider THH as a functor of rings and bimodules, and when there is no danger of confusion, we write THH(A, P ,X), even though we actually mean THH(HA, HP ,X) and so on. Whether the ring is discrete or truly simplicial is of less importance in view of the following lemma, which holds for simplicial S-algebras in general.
Lemma 1.3.1 Let A be a simplicial S-algebra, P an A-bimodule and X a space. Then there is a chain of natural pointwise equivalences
diag∗{[q]7→THH(Aq, Pq,Xq)} ≃THH(diag∗A,diag∗P ,X).
Proof: Letx∈ In+1. Using that the smash product is formed degreewise, we get that diag∗(X∧V(A, P)(x)) = X∧V(diag∗A,diag∗P)(x).
Since A and P preserve connectivity of their input, the loops in the THH-construction may be performed degreewise up to a natural chain of weak equivalences
diag∗Ω∨x(X∧V(A, P))(x))≃Ω∨x(X∧V(diag∗A,diag∗P)(x))
(see A.5.0.5 and the discussion immediately after, where the chain is described explicitly:
the map going “backwards” is simply getting rid of a redundant sin| − |). Since homotopy colimits commute with taking the diagonal, we are done.
1.3.2 Relation to Hochschild homology (over the integers)
Since, à priori, Hochschild homology is a simplicial abelian group, whereas topological Hochschild homology is aΓ-space, we could considerHH to be aΓ-space by the Eilenberg–
Mac Lane construction H: A=sAb →ΓS∗ in order to have maps between them.
We make a slight twist to make the comparison even more straight-forward. Recall the definitions of H¯: A = sAb → ΓA II.1, and the forgetful functor U: A → S∗ which is
adjoint to the free functor Z˜: S∗ → A of II.1.3.1. By definition H =UH. An¯ HZ-algebra¯ A is a monoid in (ΓA,⊗,HZ)¯ (see II.1.4.3), and is always equivalent to H¯ of a simplicial ring (II.2.2.7). As noted in the proof of Corollary II.2.2.7 the loops and homotopy colimit used to stabilize could be exchanged for their counterpart in simplicial abelian groups if the input has values in simplicial abelian groups. This makes possible the following definition (the loop space of a simplicial abelian group is a simplicial abelian group, and the homotopy colimit is performed in simplicial abelian groups with direct sums instead of wedges, see A.6.4.3):
Definition 1.3.3 Let A be an HZ-algebra,¯ P an A-bimodule, and X ∈ obΓo. Define the simplicial abelian group
HHZ(A, P ,X)q = holim−−−−−→
x∈Iq+1
Ω∨x Z[X]˜ ⊗P(Sx0)⊗ O
1≤i≤q
A(Sxi)]
!
with simplicial structure maps as for Hochschild homology. Varying X and q, this defines HHZ(A, P)∈obΓA.
Remark 1.3.4 Again (sigh), should the HZ-algebra¯ A not take flat values, we replace it functorially by one that does (for instance by replacing it to H¯ of a simplicial ring which may be assumed to be free in every degree). One instance where this is not necessary is when A = ˜ZB for some S-algebra B. Note that a ZB-module is a special case of a˜ B-module via the forgetful map U: ΓA → ΓS∗ (it is aB-module “with values in A”).
If A is a simplicial ring and P an A-bimodule, HHZ( ¯HA,HP¯ ) is clearly (pointwise) equivalent to
H(¯ HH(A, P)) ={X 7→HH(A, P ,X) = HH(A, P)⊗Z[X]}.
Definition 1.3.5 For A anHZ-algebra and¯ P an A-bimodule, there is an natural map THH(UA, UP)(X)→UHHZ(A, P)(X),
called the linearization map, given by the Hurewicz map X → Z[X]˜ and by sending the smash of simplicial abelian groups to tensor product.
In the particular case of a simplicial ring R and R-bimoduleQ, the term linearization map refers to the map
THH(HR, HR)→UHHZ( ¯HR,HQ)¯ ←∼ H(HH(R, R)).
Again, if A, P, R or Q should happen to be non-flat, we should take a functorial flat resolution, and in this case the “map” is really the one described preceded by a homotopy equivalence pointing in the wrong direction (i.e., the linearization map is then what is called a weak map).
The linearization map is generally far from being an equivalence (itisfor general reasons always two-connected). If P =A it is a cyclic map.
However, we may factor THH(UA, UP)→UHHZ(A, P) through a useful equivalence:
Lemma 1.3.6 Let A be an S-algebra, P a ZA-bimodule and˜ X ∈obΓo. The inclusion X∧P(Sx0)∧ ^
1≤i≤q
A(Sxi)→Z[X]˜ ⊗P(Sx0)⊗ O
1≤i≤q
Z[A(S˜ xi)]
induces an equivalence
THH(A, UP) −−−→∼ UHHZ( ˜ZA, P).
Proof: It is enough to prove it degreewise. If M ∈ sAb is m-connected, and Y ∈ S∗ is y-connected, then M∧Y →M ⊗Z[Y˜ ] is 2m+y+ 2connected (by induction on the cells of Y: assume Y = Sy+1, and consider M → Ωy+1(M∧Sy+1) → Ωy+1M ⊗Z[S˜ y+1]. The composite is an equivalence, and the first map is 2m + 1 connected by the Freudenthal suspension Theorem A.7.2.3). Setting M = P(Sx0) and Y = X∧V
1≤i≤qA(Sxi) we get that the map is 2x0 −2 +Pq
i=1(xi −1) + conn(X) + 2 connected, and so, after looping down the appropriate number of times,x0−q+ conn(X)connected, which goes to infinity with x0.
In the following we may not always be as pedantic as all this. We will often suppress forgetful functors, and write this as T HH(A, P) ∼ //HHZ( ˜ZA, P).
If A is an HZ-algebra and¯ P an A-bimodule, this gives a factorization THH(UA, UP) −−−→∼ UHHZ( ˜ZUA, P)→UHHZ(A, P). Remark 1.3.7 Some words of caution:
1. Note, that even if P =A, HHZ( ˜ZA, P) is not a cyclic object.
2. Note that if A is a simplicial ring, then ZHA˜ is not equal to HZA. We will discover˜ an interesting twist to this when we apply these lines of thought to additive categories instead of rings (see section 2.4).
3. In view of the equivalence HZ∧A ≃ ZA, Lemma 1.3.6 should be interpreted as a˜ change of ground ring equivalence
HHS(A, UP)≃UHHHZ(HZ∧A, P).
More generally, if k → K is a map of S-algebras, A a cofibrant k-algebra and P a K∧kA-bimodule, then
HHk(A, f∗P)≃f∗HHK(K∧kA, P) where f: A∼=k∧kA→K∧kA is the map induced by k →K.
For comparison purposes the following lemmas are important (see [225, 4.2]) Lemma 1.3.8 If A is a ring and P an A-bimodule, then there is a spectral sequence
Ep,q2 =HHp(A, πqTHH(Z, P ,X),Y)⇒πp+qTHH(A, P ,X∧Y).
Proof: For a proof, see Pirashvili and Waldhausen [225].
On a higher level, it is just the change of ground ring spectral sequence: let k →K be a map of commutative S-algebras, A a K-algebra and P a K∧kA-bimodule, and assume A and K cofibrant as k-modules, then
HHk(A, P)≃HHK(K∧kA, P)≃HHK(A,HHk(K, P))
where by abuse of notationP is regarded as a bimodule over the various algebras in question through the obvious maps.
In view of Lemma 1.3.8 we will need to know the values of π∗(THH(Z, P)) for ar-bitrary abelian groups P. These values follow from Bökstedt’s calculations in view of the isomorphism HHZ( ˜Z[Z], P)∼=HHZ( ˜Z[Z],Z)⊗P (or the equivalence THH(HZ, P)≃ THH(Z)∧HZHP) and the universal coefficient theorem:
πkTHH(Z, P)∼=
P k = 0
P/iP k = 2i−1 TorZ1(Z/iZ, P) k = 2i >0.
Lemma 1.3.9 If A is a ring and P an is A-bimodule, then the linearization map THH(A, P)→HHZ(A, P)
(and all the other variants) is a pointwise equivalence after rationalization, and also after profinite completion followed by rationalization.
Proof: In the proof of the spectral sequence of Lemma 1.3.8, we see that the edge homo-morphism is induced by the linearization map π∗THH(A, P) → π∗HH(A, P). From the calculation ofπ∗THH(Z, P)above we get that all terms in the spectral sequence above the base line are torsion groups of bounded order. Thus, πjTHH(A, P) and πjHHZ(A, P) = πjHH(A, P)differ at most by groups of this sort, and so the homotopy groups of the profi-nite completions THH(A, P)bandHHZ(A, P)bwill also differ by torsion groups of bounded order, and hence we have an equivalence THH(A, P)b(0) →HHZ(A, P)b(0).
If the reader prefers not to use the calculation of THH(Z), one can give a direct proof of the fact that the homotopy fiber of THH(A, P)→ HH(A, P) has homotopy groups of bounded order directly from the definition.
Sketch:
1. It is enough to prove the result it in each simplicial dimension.
2. As A and P are flat as abelian groups we may resolve each by free abelian groups (multiplication plays no role), and so it is enough to prove it for free abelian groups.
3. We must show that Z[X]∧˜ Z[Y˜ ]∧Z →Z[X∧Y˜ ]∧Z has homotopy fiber whose homo-topy is torsion of bounded order in a range depending on the connectivity of X, Y and Z. This follows as the homology groups of the integral Eilenberg–Mac Lane spaces are finite in a range.