Many results are most easily proven directly for ΓS∗-categories, and not by referring to a reduction to special cases. We collect a few which will be of importance.
2.5.1 THH respects equivalences
This is the first thing that we should check, so that we need not worry too much about choosing this or that model for our categories.
Lemma 2.5.2 Let F0, F1: (C, P) → (D, Q) be maps of ΓS∗-natural bimodules, and X a space. If there is a natural isomorphism η: F0 →F1, then the two maps
F0, F1: THH(C, P)(X)→THH(D, Q)(X) are homotopic.
Proof: We construct a homotopyH:THH(C, P)(X)∧∆[1]+ →THH(D, Q)(X)as follows.
If φ ∈ ∆([q],[1]) and x ∈ Iq+1 we define the map Hφ,x: V(C, P)(x) → V(D, Q)(x) by sending the c0, . . . , cq ∈ Cq+1 summand into the Fφ(0)(c0), . . . , Fφ(q)(cq) ∈ obD summand via the maps
C(c, d) −−−→ D(FF0 0(c), F0(d)) D(η
c−i,ηjd)
−−−−−→ D(Fi(c), Fj(d))
for i, j ∈ {0,1} (and P(c, d) //Q(F0(c), F0(d)) Q(η
−ic ,ηjd)
//Q(Fi(c), Fj(d))).
Clearly, the induced map Ω∨x(X∧Hφ,x) : Ω∨x(X∧V(C, P)(x))→Ω∨x(X∧V(D, Q)(x)) is functorial in x∈ Iq+1, and so defines a map Hφ: THH(C, P)(X)q → THH(D, Q)(X)q. From the construction, we see that ifψ: [p]→[q]∈∆thenψ∗Hφ=Hφψψ∗(do it separately
for ψ’s representing face and degeneracies. For the interior face maps (i.e., for 0< i < q), use that the diagram
C(ci, ci−1)∧C(ci+1, ci) −→ C(ci+1, ci−1)
yF0∧F0
yF0
D(F0(ci), F0(ci−1))∧D(F0(ci+1), F0(ci)) −→ D(F0(ci+1), F0(ci−1))
yD(η−φ(i)ci ,ηφ(i−1)ci−1 )∧D(η−φ(i+1)ci+1 ,ηφ(i)ci ) yD(η−φ(i+1)ci+1 ,ηφ(i−1)ci−1 ) D(Fφ(i)(ci), Fφ(i−1)(ci−1))∧D(Fφ(i+1)(ci+1), Fφ(i)(ci)) −→ D(Fφ(i+1)(ci+1), Fφ(i−1)(ci−1)) commutes, where the horizontal maps are composition. The extreme face maps are similar, using the bimodules P and Q).
Corollary 2.5.3 (THH respects ΓS∗-equivalences) Let C F //D beΓS∗-equivalence of ΓS∗-categories, P a D-bimodule and X a space. Then
THH(C, F∗P)(X) −−−→≃ THH(D, P)(X).
Proof: LetG be an inverse, and η: 1C
∼= //GF and ǫ: 1D
∼= //F G the natural isomor-phisms. Consider the (non commutative) diagram
THH(C, F∗P)(X) η //
F
THH(C,(F GF)∗P)(X)
F
THH(D, P)(X) ǫ //THH(D,(F G)∗P)(X)
jj G
UUUUU
UUUUUUUUUUUU
Lemma 2.5.2 then states that we get a map homotopic to the identity if we start with one of the horizontal isomorphism and go around a triangle.
Recall the notion of stable equivalences of ΓS∗-categories II.2.4.1.
Lemma 2.5.4 (THH respects stable equivalences of ΓS∗-categories) Consider a map F: (C, P)→(D, Q) of ΓS∗-natural bimodules, and assume F is a stable equivalence of ΓS∗ -categories inducing stable equivalences
P(c, c′)→Q(F(c), F(c′)) for every c, c′ ∈obC. Then F induces a pointwise equivalence
THH(C, P)→THH(D, Q).
Proof: According to Lemma II.2.4.2 we may assume that F is either a ΓS∗-equivalence, or a stable equivalence inducing an identity on the objects. If F is a ΓS∗-equivalence we are done by Corollary 2.5.3 once we notice that the conditions on P and Q imply that THH(C, P)→THH(C, F∗Q)is a pointwise equivalence.
If F is a stable equivalence inducing the identity on objects, then clearly F induces a pointwise equivalence
THH(C, P)q →THH(C, F∗Q)q →THH(D, Q)q
in every simplicial degree q.
2.5.5 A collection of other results
The approximation in Section 1.4 of THH(A) for an arbitrary S-algebra by means of the topological Hochschild homology of simplicial rings also works, mutatis mutandis, for ΓS∗-categories to give an approximation of any ΓS∗-category in terms of sAb-categories.
The proof of the following lemma is just as for S-algebras (Lemma 1.3.1)
Lemma 2.5.6 Let C be a simplicialΓS∗-category and M a C-bimodule (or in other words, {[q]7→(Cq, Mq)} is a natural bimodule). Then there is a natural pointwise equivalence
THH(diag∗C,diag∗M)≃diag∗{[q]7→THH(Cq, Mq)}. '!&"%#$..
Definition 2.5.7 Let A and B be ΓS∗-categories and M an Ao− B-bimodule. Then the upper triangular matrix ΓS∗-category
[AMB ]
is the ΓS∗-category with objectsobA ×obB and with morphism object from(a, b)to(a′, b′) given by the matrix
A(a, a′) M(a, b′) B(b, b′)
=A(a, a′)×[M(a, b′)∨ B(b, b′)]
and with obvious matrix composition as in II.1.4.4.6.
The projections from h
A(a,a′)M(a,b′) B(b,b′)
i to A(a, a′) and B(b, b′) induce S-algebra maps from [AMB ] toA and B.
Lemma 2.5.8 With the notation as in the definition, the natural projection THH([AMB ])→THH(A)×THH(B)
is a pointwise equivalence.
Proof: Exchange some products with wedges and do an explicit homotopy as in [70, 1.6.20].
For concreteness and simplicity, let’s do the analogous statement for Hochschild homol-ogy of k-algebras instead, where k is a commutative ring: let A11 and A22 be k-algebras, and let A12 be an Ao11⊗kA22-module. The group of q-simplices in HH A11A12
A22
can be written as
MOq
i=0
Ari,si
where the sum is over the set of all functions (r, s) : {0,1. . . , q} → {(11),(12),(22)}. The projection to HH(A11)⊕ HH(A22) is split by the inclusion onto the summands where r0 = . . . rq = s0 = · · · = sq. We make a simplicial homotopy showing that the non-identity composite is indeed homotopic to the non-identity. Let φ ∈ ∆([q],[1]) and y in the (r, s) summand of the Hochschild homology of the upper triangular matrices. With the convention that sq+1 =r0 we set
H(φ, y) =y, if rk =sk+1 for all k ∈φ−1(0)
and zero otherwise. We check that for j ∈ [q] we have equality djH(φ, y) = H(φdj, djy), and so we have a simplicial homotopy. Note that H(1,−) is the identity and H(0,−) is the projection (r0 = s1, . . . rq−1 = sq, rq = s0 implies that all indices are the same due to the upper triangularity).
The general result is proven by just the same method, exchanging products with wedges to use the distributivity of smash over wedge, and keeping track of the objects (this has the awkward effect that you have to talk about non-unital issues. If you want to avoid this you can obtain the general case from the Ab-case by approximating as in 1.4). Alternatively you can steal the result from I.3.6 via the equivalences
THH(C)≃HHH( ˜ZC,C)≃HHH(ZC,C) =F(C,C) to get an only slightly weaker result.
SettingM in Lemma 2.5.8 to be the trivial module you get thatTHH preserves products (or again, you may construct an explicit homotopy as in [70, 1.6.15] (replacing products with wedges). There are no added difficulties with the bimodule statement.
Corollary 2.5.9 Let C and D be ΓS∗-categories, P a C-bimodule,Q a D-bimodule. Then the canonical map is a pointwise equivalence
THH(C × D, P ×Q)→THH(C, P ,X)×THH(D, Q,X). '!&"%#$..
Recall from III.2.1.1 the canonical mapHC(S¯ 1)→SC, which in dimensionqis induced by sending the sum diagramC ∈obH(C)(q¯ +)toc∈obSqCwithcij =C{0,i+1,i+2,...,j−1,j}and obvious maps. This map factors through the (degreewise) equivalence of categories TC→ SC discussed in I.2.2.5, where TC is the simplicial category of upper triangular matrices.
Since H(C)¯ is equivalent to C×q, we get by induction (setting A=M =C and B=Tq−1C in Lemma 2.5.8) that, for eachq andX, the mapTHH( ¯H(C)(q+),X)→THH(SqC,X)is a weak equivalence. Letting q vary and using that THH can be calculated degreewise (just as in Lemma 1.3.1), we get the following corollary:
Corollary 2.5.10 LetC be an additive category and X a space. Then the mapHC(S¯ 1)→ SC induces a weak equivalence THH( ¯HC(S1),X)→THH(SC,X). '!&"%#$..
2.5.11 Cofinality
Another feature which is important is the fact that topological Hochschild homology is insensitive to cofinal inclusions (see below). Note that this is very different from the K-theory case where there is a significant difference between the K-theories of the finitely generated free and projective modules: K0f(A)→K0(A) is not always an equivalence.
Definition 2.5.12 Let C ⊆ D be a ΓS∗-full inclusion of ΓS∗-categories. We say that C is cofinal in D if for every d∈obD there exist maps
d −−−→ηd c(d) −−−→πd d such that c(d)∈obC and πdηd= 1d.
Lemma 2.5.13 Let j: C ⊂ D be an inclusion of a cofinal ΓS∗-subcategory. Let P be a D-bimodule. Then the induced map
THH(C, P)→THH(D, P) is a pointwise equivalence.
Proof: For simplicity we prove it for P =D. For eachd∈obD choose d −−−→ηd c(d) −−−→πd d,
such that ηc is the identity for all c ∈ obC. Then for every x ∈ Iq+1 we have a map V(D)(x) → V(C)(x) sending the d0, . . . , dq ∈ UDq+1 summand to the c(d0), . . . , c(dq) ∈ UCq+1 summand via
D(πd0, ηdq)(Sx0)∧. . .∧D(πdq, ηdq−1)(Sxq).
This map is compatible with the cyclic operations and hence defines a map D(π, η) : THH(D)→THH(C)
Obviously D(π, η)◦THH(j) is the identity on THH(C) and we will show that the other composite is homotopic to the identity. The desired homotopy can be expressed as follows.
Let φ∈∆([q],[1]) and let d η
id
−−−→ ci(d) π
di
−−−→ d be
(
d −−−→ηd c(d) −−−→πd d if i= 1 d=d=d if i= 0
The homotopy THH(D)∧∆[1]+→THH(D) is given by Hφ,x: V(D)(x)→V(D)(x) send-ing thed0, . . . , dq ∈obUDq+1 summand to thecφ(0)(d0), . . . , cφ(q)(dq)∈obUDq+1 summand via
D(πdφ(0)0 , ηdφ(q)q )(Sx0)∧. . .∧D(πdφ(q)q , ηφ(q−1)dq−1 )(Sxq).
2.5.14 Morita invariance
If A is an S-algebra, let FA be the ΓS∗ category whose objects are the natural numbers, with n thought of as the free A-module of rank n, and FA(m, n) = S∗(m+, n+∧A), the n×m matrices with coefficients in A as in II.1.4.4.6. Let FAk be the full subcategory of objects of rank less than or equal to the natural number k.
This should be compared with the situation when R is a discrete ring. Then FRk is the Ab-category with objects the natural numbers less than or equal tokand a morphism from m to n is an n×m-matrix (in the usual sense) with entries in R. By sending wedges to products, we see that the ΓS∗-category FfRk associated with FRk (by taking the Eilenberg-Mac Lane construction on all morphism spaces, c.f. II.1.6.2.2) is stably equivalent toFHRk , and so THH(FHRk )→∼ THH(FfRk).
Thinking of the S-algebra MatkA as the full subcategory of FA whose only object is k, we get a cofinal inclusion MatkA ⊆ FAk: for n ∈ obFAk, the maps ηn and πn of Definition 2.5.12 ensuring cofinality are given by the matrices representing inclusion into and projection onto the first n coordinates. Hence we get that
Lemma 2.5.15 LetAbe anS-algebra andP anA-bimodule. Then the inclusionMatkA→ FAk induces a pointwise equivalence THH(MatkA,MatkP)→∼ THH(FAk, P). '!&"%#$..
Here we have writtenP also for the FAk-bimodule given byP(m, n) = S∗(m+, n+∧P)with matrix multiplication from left and right by matrices with entries in A.
The inclusion of the rank one A-module, A⊆ FAk, is not cofinal, unless k = 1, but still induces an equivalence:
Lemma 2.5.16 Let A be anS-algebra and P an A-bimodule. Then the inclusionA⊆ FAk induces a pointwise equivalence
THH(A, P)→∼ THH(FAk, P).
Proof: By 1.4, 1.3.1 and 1.3.8 it is enough to prove the lemma for Hochschild homology of discrete rings (alternatively, you must work with homotopies not respecting degeneracies as in [70]).
For the rest of the proof A will be a discrete ring and P an A-bimodule. It is helpful to write out HH(FAk, P)q by means of distributivity as
M
(n,r,s)
P ⊗A⊗q
where the sum is over the tuples n = (n0, . . . , nq), r = (r0, . . . , rq) and s = (s0, . . . , sq) of natural numbers where ri, si ≤ ni ≤k for i= 0, . . . q. The isomorphism to HH(FAk, P)q is given by sendingp⊗a1⊗· · ·⊗aqin the(n,r,s)-summand to(p prsqinr0)⊗Nq
i=1(aiprsi−1inri) in then∈(obFAk)×q+1summand. Hereiniandprirepresent theith injection and projection matrices.
There is a map tr: HH(Fk, P) → HH(A, P) sending a = p⊗a1 ⊗ · · · ⊗aq in the (n,r,s)-summand to
tr(a(n,r,s)) =
(a if rk =sk for all k ∈[q]
0 otherwise.
Clearly, the composite HH(A, P) → HH(FAk, P) → HH(A, P) is the identity, and we are done if we can construct a concrete simplicial homotopy H: HH(FAk, P)⊗Z[∆[1]]˜ → HH(FAk, P) between the other composite and the identity.
If φ: [q]→[1]∈∆, and (n,r,s) is a tuple as above, let (nφ,rφ,sφ)be the tuple where the ith factor in each of the three entries is unchanged if φ(i) = 1 and set to 1if φ(i) = 0.
Then we define
H(a(n,r,s)⊗φ) =
(a(nφ,rφ,sφ) if rk =sk for all k∈φ−1(0).
0 otherwise.
A direct check reveals that this defines the desired simplicial homotopy.
Remark 2.5.17 We noted earlier that in our presentation there was an unfortunate lack of a map THH(A)→THH(Matk(A)) realizing Morita invariance. The natural substitute is THH(A)→∼ THH(FAk)←∼ THH(Matk(A)).
Since topological Hochschild homology commutes with filtered colimits (loops respect filtered colimits (A.1.5.5) andV(FA, P)(x) = limk→∞V(FAk, P)(x)for all x∈ Iq+1) we get the following corollary:
Corollary 2.5.18 LetAbe anS-algebra andP anA-bimodule. Then the inclusion ofAas the rank one module inFAinduces a pointwise equivalenceTHH(A, P)→THH(FA, P). '!&"%#$..
2.5.19 Application to the case of discrete rings
As an easy application, we will show how these theorems can be used to analyze the topological Hochschild homology of a discrete ring.
For a discrete ring A recall the category PA of finitely generated projective modules (I.2.1.3) and the category FA of finitely generated free modules (I.2.1.4). Again, if P is an A-bimodule, we also write P for the PA-bimoduleHomA(−,− ⊗AP) ∼= PA(−,−)⊗A
P: PA× PAo →Ab.
By sending wedges to products, we see that the ΓS∗-category associated with FA (by taking the Eilenberg-Mac Lane construction on all morphism spaces to achieve what would be recorded as FeA) is stably equivalent to the ΓS∗-category FHA of finitely generated free HA-modules, and so the results for the latter found in the previous section give exactly the same results for the former. In particular, THH(A, P)→THH(FA, P) is a pointwise equivalence. Here we have again used the shorthand of writingTHH(A, P)when we really mean THH(HA, HP), and likewise for THH(FA, P).
Since FA⊆ PA is a cofinal inclusion we get by Lemma 2.5.13 that
Lemma 2.5.20 Let A be a discrete ring, and let P be an A-bimodule. Then the inclusion FA⊆ PA induces a pointwise equivalence
THH(FA, P) −−−→∼ THH(PA, P). '!&"%#$..
Collecting the results, we get
Theorem 2.5.21 The (full and faithful) inclusion of A in PA as the rank 1 free module induces a pointwise equivalence
THH(A, P) −−−→∼ THH(PA, P). '!&"%#$..