Table Of ContentCyclic Homology Handout Dave Penneys
Quantum Geometry Seminar 4/28/08
Most of the following information comes from Loday’s Cyclic Homology.
I. The Cyclic, Simplicial, and Semi-Simplicial Categories
The Cyclic Category. The category c∆ has the following presentation:
Objects: [n] = {0 < 1 < ··· < n} for n ∈ N.
Morphisms: composites of
δ : [n] −→ [n+1] for 0 ≤ i ≤ n+1,
i
σ : [n−1] −→ [n] for 0 ≤ j ≤ n, and
j
τ: [n] −→ [n]
such that the following relations hold:
(i) δ δ = δ δ for i < j,
j i i j−1
(ii) σ σ = σ σ for i ≤ j,
j i i j+1
δ σ for i < j
i j−1
(iii) σ δ = id for i = j,j +1
j i [n]
δ σ for i > j +1,
i−1 j
(iv) τn+1 = id ,
[n]
(
δ τ for 1 ≤ i ≤ n
i−1
(v) τδ =
i
δ for i = 0,
n
(
σ τ for 1 ≤ i ≤ n
i−1
(vi) τσ =
i σ τ2 for i = 0.
n
We can take δ to be the strictly increasing injection that skips i, σ the increasing surjection such
i j
that j,j+1 7→ j, and τ the cyclic permutation such that τ(k) = k+1 mod (n+1) for 0 ≤ k ≤ n.
The Simplicial Category. The category s∆ is the subcategory of c∆ generated by the δ ’s
i
and the σ ’s.
j
The Semi-Simplicial Category. The category ss∆ is the subcategory of c∆ (or s∆) gener-
ated by the δ ’s.
i
II. Cyclic, Simplicial, and Semi-Simplicial Objects
Definition: A cyclic (respectively simplicial, semi-simplicial) object in a category A is a functor
X : c∆op → A
•
(respectively s∆op, ss∆op). Denote the category of cyclic (respectively simplicial, semisimplicial)
A-objects by cA (respectively sA, ssA), and note that cA = Fun(c∆op,A), the category of functors
from c∆op to A (respectively sA = Fun(s∆op,A), ssA = Fun(ss∆op,A)).
1
Usually we write X = X ([n]), d = X (δ ), s = X (σ ), and t = X (τ). Note that
n • i • i j • j •
d : X −→ X for 0 ≤ i ≤ n, n > 1
i n n−1
s : X −→ X for 0 ≤ j ≤ n, and
j n n+1
t: X −→ X .
n n
We have the following relations among these maps:
(i) d d = d d for i < j,
i j j−1 i
(ii) s s = s s for i ≤ j,
i j j+1 i
s d if i < j
j−1 i
(iii) d s = id if i = j,j +1
i j
s d if i > j +1
j i−1
(iv) tn+1 = id,
n
(
td for 1 ≤ i ≤ n
i−1
(v) d t =
i
d for i = 0,
n
(
ts for 1 ≤ i ≤ n
i−1
(vi) s t =
i t2s for i = 0.
n
From this point on, k will be a commutative ring. Note that if C : c∆op → MOD, the category of
• k
left k-modules, we set t = (−1)nC (τ) to satisfy Loday’s sign convention, which accounts for the
•
sign of the cyclic permutation in S . Hence (v) and (vi) above are replaced with:
n+1
(
−td for 1 ≤ i ≤ n
i−1
(v) d t = and
i (−1)nd for i = 0
n
(
−ts for 1 ≤ i ≤ n
i−1
(vi) s t =
i (−1)nt2s for i = 0.
n
III. Various Categories and Functors
Given a category A, we will write a ∈ A to mean a ∈ Ob(A) and f ∈ A(a,b) if f: a → b for a,b ∈ A.
We already know about the categories ss∆, s∆, and c∆; ssA, sA, and cA for a given category
A; and the category Fun(A,B) for given categories A and B.
Definition: Categories will be denoted by the sans-serif font: ABC, etc.
(1) Grpisthecategoryofgroups,andABisthecategoryofabeliangroups,whichwillsometimes
be referred to as Z-modules since the term “cyclic abelian group” would be ambiguous.
(2) Set is category of sets.
(3) Top is the category of topological spaces with continuous maps.
(4) MOD (respectively MOD ) is the category of left (respectively right) k-modules.
k k
(5) CPLX will denote the category of chain complexes of abelian groups, and CPLX (respec-
k
tively CPLX ) will denote the category of chain complexes in MOD (respectively MOD ).
k k k
(6) ALG is the category of k-algebras.
k
(7) Cat is the category of small categories.
Sometimes we add a subscript to the category to denote certain properties.
2
Definition: U will denote the forgetful functor. We have obvious forgetful functors
oo U oo U
ssA sA cA.
In fact, there are left adjoints to these functors.
Definition: The functor F = FREE: Set → AB is given by X 7→ ZhXi, the free abelian group
on the elements of X.
F extends to a functor F: ssSet → ssAB by F(X ) = ZhX i and extending the maps Z-linearly.
n n
Similarly, we can look at the functor F: ssSet → ss MOD. We can also replace ss with s or c in the
k
above discussion.
F is left adjoint to U: AB → Set.
Definition: The functor ALT: ss MOD → CPLX is given by C 7→ (C ,b) where C is the
k k • ∗ n
same as before, and b is the alternating sum of the d ’s:
i
n
X
b = (−1)id .
i
i=0
Once again, we can replace ss with s or c in the above discussion.
IV. Examples
(1) Singular Homology. Let X be a topological space, and let ∆n = co{e ,...,e } ⊂ Rn+1
0 n
be the standard n-simplex. For 0 ≤ i ≤ n+1, define
δ : ∆n −→ ∆n+1 by
i
n n
X X
α e 7−→ α e
j j j δi(j)
j=0 j=0
where α ,...,α ∈ [0,1] with α +···+α = 1. Define the semi-simplicial set
0 n 0 n
∆ (X): ss∆ → SET
•
by S (X) = {continuous f: ∆n → X} and d = ∆ (δ ) = δ∗, i.e. d (f) = f ◦δ :
n i • i i i i
X oo f ∆n+1 oo δi ∆n.
Applying the functor F: ssSET → ssAB, we have a semi-simplicial Z-module S (X) =
•
F(∆ (X)). Now applying the functor ALT: ssAB → CPLX and the functor H : CPLX →
• n
AB gives the nth singular homology of X. Hence, singular homology is the composite of the
following functors:
AB oo Hn CPLX oo ALTssAB oo F ssSET oo ƥ Top.
3
(2) Group Homology. Let A be a small category.
Definition: The nerve of A is the simplicial set N (A) is given by
•
N (A) = Ob(A),
0
(cid:26) (cid:12) (cid:27)
oo f (cid:12)
N1(A) = b a (cid:12)a,b ∈ A ,
(cid:12)
(cid:26) (cid:12) (cid:27)
oo g oo f (cid:12)
N2(A) = c b a (cid:12)a,b,c ∈ A ,
(cid:12)
and so forth. The maps d are given by delting the ith object, e.g.
i
(cid:18) (cid:19)
oo g oo f oo g
d0 c b a = c b ,
(cid:18) (cid:19)
oo g oo f oo g◦f
d1 c b a = c a , and
(cid:18) (cid:19)
oo g oo f oo f
d2 c b a = b a .
The maps s are given by adding the identity morphism for the jth object, e.g.
j
(cid:18) (cid:19)
s0 c oo g b oo f a = c oo g b oo f a oo ida a ,
(cid:18) (cid:19)
s1 c oo g b oo f a = c oo g b oo idb b oo f a , and
(cid:18) (cid:19)
s2 c oo g b oo f a = c oo idc c oo g b oo f a .
If A has only one object, Ob(A) = {∗}, then N (A) is the cyclic set where the map t is given
•
by the cyclic permutation, e.g.
(cid:18) (cid:19)
oo h oo g oo f oo g oo f oo h
t ∗ ∗ ∗ ∗ = ∗ ∗ ∗ ∗ .
Suppose now that A is a group, i.e. Ob(A) = {∗} and all morphisms are invertible. Then
applying the functors U, F, ALT, and H , we get group homology. That is, if G = N (A),
n 1
then H (G) is given by the following composite of functors:
n
AB oo Hn CPLX oo ALT cAB oo F cSET oo N• Cat oo Grp,
∗
where Cat is the category of all small categories with one object.
∗
(3) Hochschild Homology. Let A be a unital k-algebra, and let M be an A−A-bimodule.
DefinethesimplicialmoduleC (A,M)byC (A,M) = M⊗A⊗n where⊗means⊗ . Define
• n k
the d ’s by
i
d (a ⊗a ⊗a ⊗···⊗a ) = a a ⊗a ⊗···⊗a ,
0 0 1 2 n 0 1 2 n
d (a ⊗···⊗a ⊗a ⊗···⊗a ) = a ⊗···⊗a a ⊗···⊗a for 0 < i < n, and
i 0 i i+1 n 0 i i+1 n
d (a ⊗a ⊗···⊗a ⊗a ) = a a ⊗a ⊗···⊗a
n 0 1 n−1 n n 0 1 n−1
for a ∈ M and a ,...,a ∈ A. Define the s ’s by
0 1 n i
s (a ⊗···⊗a ⊗a ⊗···⊗a ) = a ⊗···⊗a ⊗1⊗a ⊗···⊗a
i 0 i i+1 n 0 i i+1 n
4
(where 1 ∈ A). If M = A, then C (A,A) is a cyclic module where t is given by
•
t(a ⊗···⊗a ⊗a ) = (−1)na ⊗a ⊗···⊗a .
0 n−1 n n 0 n−1
TakingALT andH givesthenth HochschildhomologyH (A,M), soHochschildhomology
n n
is a composite of the following functors:
AB oo Hn CPLX oo ALT s MOD oo C•(−,M) ALG
k k k unital
If M = A, we write HH (A) = H (A,A), and the above sequence is altered by replacing
n n
s MOD with c MOD.
k k
If G: ALG → A is a functor where A is an abelian category, we can extend G to
k unital
a functor Ge: kALG → A. For I ∈ kALG, we can form its unitalization Ie= k⊕I, and define
Ge(I) = coker(G(i)), where i: k → Iein the natural inclusion:
oooo oo p oo i oo
0 I Ie k 0.
Hence, Hochschild homology extends to a functor ALG → AB.
k
V. Cyclic Homology
Given a cyclic module C , there is a classical complex arising from calculations in group homology:
•
oo 1−t oo N oo 1−t oo N
··· C C C ···
n n n
where N = 1+t+t2 +···+tn is the “norm map.” A group action on a space X is a continuous
map
G×X −→ X.
We can think of this classical complex “acting” on (C ,b) = ALT(C ) to get the cyclic bicomplex
∗ •
CC :
∗∗
b −b0 b −b0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo 1−t oo N oo 1−t oo N
C C C C
2 2 2 2
b −b0 b −b0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo 1−t oo N oo 1−t oo N
C C C C
1 1 1 1
b −b0 b −b0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo 1−t oo N oo 1−t oo N
C C C C
0 0 0 0
where b0 = d −d +···+(−1)n−1d . The relations of a cyclic module imply b(1−t) = (1−t)b0
0 1 n−1
and b0N = Nb, so the bicomplex above is well defined.
Definition: Then nth cyclic homology of a cyclic module C is the nth total homology of the
•
associated cyclic bicomplex CC :
∗∗
HC (C ) = H (Tot (CC )).
n • n ∗ ∗∗
Cyclic homology is thus the composite of the following functors:
AB oo Hn BCPLX oo c MOD
k k
5
We have an additional method of taking homology of a cyclic module by the functors described
before, which will be denoted by HH :
n
HH (C ) = H (ALT(C )) = H (C ,b).
n • n • n ∗
From this bicomplex construction, we can reconstruct Connes’ initial construction of cyclic homol-
ogy and many of the tools he developed for calculation. First, note that the complex (C ,−b0) is
∗
contractible via contraction −s = −(−1)nts . We will need the following lemma:
n
Killing Contractible Complexes Lemma: Let (A ⊕A0,d) ∈ CPLX
∗ ∗ k
"α β#
d=
γ δ
··· oo A ⊕A0 oo A ⊕A0 oo ···
n−1 n−1 n n
such that (A0,δ) ∈ CPLX is contractible via contraction h. Then (A ,α−βhγ) ∈ CPLX, and the
∗ k ∗ k
inclusion
(id,−hγ): (A ,α−βhγ) −→ (A ⊕A0,d)
∗ ∗ ∗
is a quasi-isomorphism.
(1) Connes’ Periodicity Exact Sequence. Let CC be the cyclic bicomplex associated to
∗∗
(2)
the cyclic module C . Let CC be the bicomplex obtained from CC by looking at only
• ∗∗ ∗∗
the first two columns, and let CC[2] be the bicomplex obtained from CC by shifting
∗∗ ∗∗
the bicomplex two columns to the right. We then get an exact sequence of bicomplexes
0 oo CC[2]∗∗ oo CC∗∗ oo CC∗(2∗) oo 0
which yields an exact sequence in total complexes, which in turn, yields a long exact se-
quence in homology via the snake lemma:
oo oo oo oo oo oo
··· HH (C ) HC (C ) HC (C ) HH (C ) HC (C ) ··· .
n−1 • n−2 • n • n • n−1 •
Note that
(2) ∼
H (Tot (CC )) = HH (C ) = H (C ,b)
n ∗ ∗∗ n • n ∗
by the lemma, and clearly
H (Tot (CC[2] )) = HC (C ).
n ∗ ∗∗ n−2 •
(2) Connes’ Bicomplex. By applying the lemma to the −b0 columns of the cyclic bicomplex
which are contractible via −s, we get Connes’ bicomplex BC where B = (1−t)sN:
∗∗
b b b
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo B oo B
C C C
2 1 0
b b
(cid:15)(cid:15) (cid:15)(cid:15)
oo B
C C
1 0
b
(cid:15)(cid:15)
C
0
6
(3) Connes’Complex. Connes’initialdefinitionofcyclichomologywasgivenasthehomology
of a complex instead of a total complex of a bicomplex. Since b(1−t) = (1−t)b0, we have
that b descends to a map
··· oo eb Cλ oo eb Cλ oo eb Cλ oo eb ···
n−1 n n+1
where Cnλ = Cn/im(1−t). Define Hnλ(C•) = Hn(C∗λ,eb).
Theorem: If Q ⊆ k, there is a canonical isomorphism Hλ(C ) ∼= HC (C ).
n • n •
Proof. Let
n
1 −1 X
h0 = id, h = iti
n+1 n+1
i=1
be maps C → C :
n n
oo (1−t) oo N oo (1−t) oo N
Cn NNNNNNNhNNNNNN&& C(cid:15)(cid:15)nidNNNNNNNhN0NNNNN&& C(cid:15)(cid:15)nidNNNNNNNhNNNNNN&& C(cid:15)(cid:15)nidNNNNNNNhN0NNNNN&& ···
oo (1−t) oo N oo (1−t) oo N
C C C C ···
n n n n
From the relations of a cyclic module, h0N+(1−t)h = id and Nh0+h(1−t) = id. Hence, the rows
of CC are acyclic augmented complexes with H = Cλ, and there is a first quadrant spectral
∗∗ 0 n
sequence IIE0 ⇒ HC (C ) with
p,q p+q •
IIE0 = CC and d0 = d ,
p,q p,q h
IIE1 = H(CC ,dh) = δ Cλ and d1 = [dv], and
p,q p,q p,0 q
IIEp2,q =II Ep∞,q = H(δp,0Cqλ,[dv]) = δp,0Hq(C∗λ,eb) and d2 = 0.
(cid:3)
VI. Elementary Computations
(1) Hochschild Homology. Let A ∈ ALG and M ∈ MOD . Clearly
k unital A A
H (A,M) = M = M/ham−mai,
0 A
the module of coinvariants of M by A. If M = A, then HH (A) = A/[A,A]. If A is
0
commutative, HH (A) = A. If A = k, we have the cylic module C (k) given by C (k) =
0 • n
k⊗(n+1) ∼= k, and each d is the identity. Hence, the Hochschild complex C (k) is given by
i ∗
oo 0 oo 1 oo 0 oo 1 oo
k k k k k ···
and we have HH (k) = k and HH (k) = 0 for all n > 0.
0 n
7
(2) Cyclic Homology. We have C (A) = A⊗n+1 for A ∈ ALG , and Connes’ boundary
n k unital
map B: A⊗n+1 −→ A⊗n+2 is given explicitly by
n
X
B(a ⊗···⊗a ) = (1−t)(−1)nts ti(a ⊗···⊗a )
0 n n 0 n
i=0
n
X
= (1−t)(−1)nts (−1)ni(a ⊗···⊗a ⊗a ⊗···⊗a )
n i n 0 i−1
i=0
n
X
= (1−t)(−1)nt (−1)ni(a ⊗···⊗a ⊗a ⊗···⊗a ⊗1)
i n 0 i−1
i=0
n
X
= (1−t)(−1)n(−1)n+1 (−1)ni(1⊗a ⊗···⊗a ⊗a ⊗···⊗a )
i n 0 i−1
i=0
n
X
= (−1)ni+1(1⊗a ⊗···⊗a ⊗a ⊗···⊗a )
i n 0 i−1
i=0
+(−1)n(i+1)+1(a ⊗1⊗a ⊗···⊗a ⊗a ⊗···⊗a ).
i−1 i n 0 i−2
It is clear that HC (A) = HH (A). If A = k, Connes’ bicomplex is given by
0 0
1 0 1 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo −6 oo 0 oo −2
k k k k
0 1 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
oo 0 oo −2
k k k
1 0
(cid:15)(cid:15) (cid:15)(cid:15)
oo −2
k k
0
(cid:15)(cid:15)
k
Note that B: C → C is multiplication by −2(2n + 1) and B: C → C is
2n 2n+1 2n+1 2n+2
the zero map. Letting dodd denote d: Tot (BC ) −→ Tot (BC ) and deven denote
2n+1 ∗∗ 2n ∗∗
d: Tot (BC ) −→ Tot (BC ), it is clear that dodd = 0 and deven is onto with kernel
2n ∗∗ 2n−1 ∗∗
isomorphic to k. Hence
(
k for n even
∼
HC (k) =
n
0 for n odd.
Theorem: Hochschild and cyclic homology are Morita invariant, i.e. if A and B are Morita equiv-
∼ ∼
alent k-algberas, then HH (A) = HH (B) and HC (A) = HC (B).
n n n n
Corollary: HH (M (k)) = k and HH (k) = 0 for all n > 0, and
0 n n
(
k for n even
HC (M (k)) =
n n
0 for n odd.
8
