Higher Order Hochschild Homology and Its
Decompositions
Kristine Bauer
February 8, 2002
Abstract
Let k be a field of characteristic 0, A a k-algebra and M an A-
module. In this paper we see to provide a decomposition of FA O (S1 ^
Y ) where S1 ^ Y is a (suspension) simplicial finite pointed set and FA
is a functor from the category of finite pointed sets to k-vector spaces
taking the finite pointed set [[n]] = {0, 1, . .,.n} to M A n . To do
this, we show that FA O (S1 ^ Y ) is a Hopf algebra under maps whose
existence is suggested by the pinch and fold maps on the circle, S1.
We are then able to apply the methods which Loday and Gerstenhaber
and Schack used to obtain a decomposition of Hochschild homology,
FA O S1. Finally, we show that this decomposition recovers the re-
cent decomposition of higher order Hochschild homology, FA O Sn by
Pirashvili.
1 Introduction
Our main result is a rational decomposition of a generalization of higher or-
der Hochschild homology. This result recaptures two similar known results.
The first such result is that of Loday [L2 ] and Gerstenhaber and Schack
[G-S ], who independently discovered a decomposition of Hochschild homol-
ogy. They accomplished this by showing that the reduced Hochschild chain
complex is a differential graded Hopf algebra, and by then using the eulerian
idempotents associated to this Hopf algebra to obtain the decomposition.
1
Later, McCarthy was able to give a geometric interpretation to this decom-
position [McC ] using the edgewise subdivision of a simplicial set and the
r-fold covering map of the circle.
More recently, Pirashvili has extended this decomposition to a new chain
complex and homology, called higher Hochschild homology [P ]. Let k be a
field of characteristic zero. One can think of Hochschild homology as the
homology of the chain complex associated to the simplicial vector space
__S1_// __FA_//
op F in* V ect=k
where S1 is the simplicial circle and FA is the functor which takes a finite
pointed set of cardinality n to M A n for any commutative k algebra
A and A-module M (see x4 for a precise definition). Using this way of
looking at Hochschild homology, Pirashvili extended the definition to "n-
th higher order Hochschild homology", the homology of the chain complex
associated to the simplicial vector space FA O Sn. Furthermore, he obtained
a decomposition of higher Hochshild homology which recovers the previously
known decomposition of Hochschild homology in the case n = 1.
The goal of this paper is to study functors of the form FA O X for any
simplicial finite pointed set X. We obtain a decomposition of this functor
for any simplicial finite poined set X of the form S1 ^ Y which recovers the
two decompositions just mentioned. Following the methods employed by Mc-
Carthy [McC ] to give a geometric description of the decomposition of HH*,
we use the edgewise subdivision of a simplicial set to provide a simplicial
version of the usual pinch map on a circle. We then show that the pinch map
and the fold map on the S1 coordinate of S1 ^ Y induce a differential graded
Hopf algebra structure on the chain complex associated to FA O (S1 ^ Y ).
Using this structure, we are able to proceed in a manner analogous to the
proofs of the original decompositions of Loday and Gerstenhaber and Schack.
Finally, we are able to show that in the case Y = Sn-1, the image of the
r-fold covering map on Sn (induced by the r-fold covering map on the first
S1-coordinate) is the same on the summands of this new decomposition as
well as the summands of the decomposition provided by Pirashvili. Hence
this recovers Pirashvili's decomposition.
Remark 1.1. The key point in obtaining our decomposition of higher order
Hochschild homology is that the chain complex FA O Sn is actually a differen-
tial graded Hopf algebra. This has lead to joint work with Randy McCarthy
2
in which we are able to show that functors from pointed categories to the
category of k-modules (for k a field containing Q) split into a product of the
layers of their Goodwillie towers when evaluated on objects which resemble
Hopf algebras (i.e. objects which have a comultiplication map with respect
to the fold map as multiplication) whenever the Goodwillie tower converges
on such an object. One can reinterpret the results obtained here in terms of
Goodwillie calculus by composing the forgetful functor, U, from differential
graded commutative algebras to k-modules with FA O (S1 ^ Y ). Applying the
new result, one then obtains the current decomposition as the layers of the
Goodwillie tower of U evaluated on the object FA O (S1 ^ X). Furthermore,
the Goodwillie tower of U is well understood and the layers can be expressed
in a particularly nice way [KM ]. Using this, one concludes that when n is
even,
Dm U(FA (Sn)) = (n-1)m m D1U(FA (S1))
and when n is odd,
Dm U(FA (Sn)) = (n-1)mSm D1U(FA (S1))
where m is the degree m part of the exterior algebra and Sm is the degree
m part of the symmetric algebra. In other words, the layers of higher order
Hochschild homology rely solely on the layers of Hochschild homology. The
details of this interpretation will appear in a subsequent paper with Randy
McCarthy.
This work was completed during my graduate study at the University
of Illinois. I am indebted to my advisor, Randy McCarthy, without whose
insights and encouragement this would not have been possible. During the
preparation of this paper I was partially supported by the Clay Mathematics
Institute.
2 L-realizations of a functor F
The main purpose of this section is to rewrite the geometric realization of
F O L, where L is a simplicial set, in terms of powers of |L|. Later, we will
see that the summands of the decomposition of HHL*(A) are identified by
the image of the map induced by covering maps of |L|. The image of this
map varies with its relation to the image of the covering map on powers of
3
|L|. In particular, when |L is a circle, the r-fold covering map acts by rn on
the n-th summand.
Let Sets* be the category whose objects are basepointed sets and whose
morphisms are all basepoint preserving maps, and let Fin * be the sub-
category of finite pointed sets. Abusing notation, we will denote objects
of Fin* by [[n]]= {*, 1, . .,.n}, where * is the basepoint of each set (techni-
cally, this is a skeletally small sub-category of the category of finite pointed
sets). Let be the standard simplicial category. Its objects are also finite
sets denoted by [n] = {0, 1, . .,.n} and its morphisms are order preserving
monotonic maps. Let L be a simplicial finite pointed set, i.e., a contravariant
functor L : ! Fin*, and F : Fin* ! Sets* be a covariant functor. The
composition F OL is a pointed simplicial set, and one can form the usual geo-
metric realization, |F OL|. Define another realization, called the L-realization
of F , to be the set
a
|F |L = F ([[n]]) x |HomFin* ([[n]], L(-))|=~ .
n2N
The relation ~ is given as follows: if f : [[n]]! [[m]]is a morphism in
Fin*, and if we let f*u denote F (f)(u) for u 2 F ([[n]]) and f*(ff) = ff O f f*
*or
ff 2 HomFin* ([[m]], L(-)), then (f*u, ff)~ (u, f*ff).
P l
Let l = {(t0, . .,.tl) | ti 2 R s.t. i=0ti= 1} be the l-simplex and let
g be a morphism from [l] to [k] in . Define g*fl := L(g) O fl for fl 2
HomFin*([[n]], L([k])). For t = (t0, . .,.tl) 2 l define g*t = (u0, . .,.uk) by
the formula 8
< 0 g-1(j) = ;
uj = P -1
: tffk g (j) = ff.
ffk2ff
where ff = {ffi} is a subset of {0, 1, . .,.l}. Then the usual geometric real-
ization is given by
a
|HomFin* ([[n]], L(-))| = HomFin* ([[n]], L([l])) x l=~
l2N
where here ~ is given by (g*fl,`t) = (fl, g*t). Notice that by expanding
|HomFin* ([[n]], L(-))| into l2NHomFin* ([[n]], L([l])) x l=~ , we may write
an element of the set |F |L as a triple (u, ff, t), where u 2 F ([[n]]), ff 2
HomFin*([[n]], L([l])), and t 2 l. On the other hand, an element of |F O L| =
4
` m
m2N F (L([m])) x =~ is a pair (v, s). An explicit homeomorphism from
|F |L to |F O L| is given by
oe(u, ff, t) = (F (ff)(u), t).
One can check that this is well defined (preserves the equivalence relation)
and has a well defined inverse oe-1 : |F O L| ! |F |L for (v, s) 2 F (L([l])) x*
* l
given by
oe-1(v, s) = (v, id, s) 2 F ([[n]]) x HomFin* ([[n]], L([l])) x l
However, there is a more concise way of seeing this fact by viewing |F |L as
a certain coend. We may write |F |L as a tensor product of functors with
equivalences following from the properties of tensors and Yoneda's Lemma:
F (-) Fin*|HomFin* (-, L(-))| = F (-) Fin*HomFin* (-, L(-)) (-)
= (F O L)(-) (-)
and (F O L) (-) is |F O L|.
Either way, we have the following:
Lemma 2.1. |F |L ~=|F O L|.
We would now like to look at some of the properties of |F |L. First note
that the construction of |F |L is suitably natural in L. Let T : L ! L0 be a
natural transformation_of pointed simplicial sets. Then T_induces_a natural
transformation T F : |F |L ! |F |L0 for each functor F . T is given by:
__
TF : (u, ff) ! (u, T O fi)
for u 2 F ([[n]]) and fi 2 HomFin* ([[n]], L(-)). This is well defined on equiv-
alence classes, since for f 2 HomFin* ([[n]], [[m]]), ff 2 HomFin* ([[m]], L(-))
and u as above with (f*u, ff)~ (u, f*ff) , then
__ * *
TF(f u, ff) = (f u, T O ff)
__
TF (u, f*ff)= (u, T O f*ff) = (u, f*T O ff)
5
This induces a commutative diagram:
|F |L_______//_|F |L0
| |
| |
fflffl| fflffl|
|F O L|_____//|F O L0|.
One can also check that the construction is natural in F .
It isWpossible to express |HomFin* ([[n]], L)| in terms of |L|. Note that si*
*nce
[[n]]= [[1]], we have
n
HomFin* ([[1]], L)xn ~= HomFin*([[n]], L).
If ff 2 HomFin* ([[n]], L), we can make an explicit identification
ff 7! ff1 x . .x.ffn (1)
where ffi is defined by the property that ffi(1) = ff(i). We also have an
isomorphism HomFin* ([[1]], L) ' L which is given by identifying a map in
HomFin*([[1]], L) with the image of 1, an element of L.
Lemma 2.2. There is an equivalence of topological spaces
a
|F |L = F ([[n]]) x |L|xn =~
n2N
Proof.The proof is now a series of equivalences:
a
|F |L = F ([[n]]) x |HomFin* ([[n]], L)|=~
n2N
a
' F ([[n]]) x |HomFin* ([[1]], L)xn |=~
n2N
a
' F ([[n]]) x |Lxn |=~
n2N
a
' F ([[n]]) x |L|xn =~
n2N
Also note that one can describe the equivalence relations at each stage by
identifying the ff with (ff1, . .,.ffn) as in Equation 1 and by then identifying
the ffi with elements of ffi(1) of L. This also makes it easy to check that
__
these equivalence relations are preserved. |__|
6
3 Operators on |F |L
We want to define operators on |F OL|. In particular, we will be interested in
finding a way to operate on |F O S1| by the r-fold covering map of the circle.
One way to obtain the r-fold cover is to first "pinch" the circle into a bouquet
of r-circles, and then fold these circle together again. The fold map exists by
the universal property of the coproduct. However, the "pinch" map is not a
simplicial map. One can make a simplicial pinch map by approximating the
circle with one which has r 0-cells instead of 1 0-cell, and then identifying
those to one point. To do this, we will follow the construction in [McC ] which
utilizes edgewise subdivisions of simplicial sets. We review the construction
here for completeness.
Let sdr: ( )xr ! be the functor given on ([n1], . .,.[nr]) by
([n1], . .,.[nr]) ! [n1] * . .*.[nr]
where [n1] * . .*.[nr] is called the concatenation of [n1] through [nr]. The
concatenation is the object of obtained by "stacking" the sets [ni] next to
each other to obtain a new ordered set [G ]. When n1 = . .=.nr, then we
produce a functor from ! via the diagonal functor from ! xr. We
define
sdr([n]) = [n] * . .*.[n].
Unless otherwise indicated, we will mean sdr to be a functor of one variable
by using the diagonal this way. Then the r-th edgewise subdivision of L
is the composition of functorsPL O sdr, which is written sdrL. If we write
n-1 = {(t1, . .,.tn)| iti = 1} as in Section 2 then we have a map dr :
n-1 ! rn-1 given by
t = (t1, . .,.tn) ! (t=r, . .,.t=r).
We have the following lemma from [G ] and [BHM ]:
Lemma 3.1 ([G ] and [BHM ]). The map 1 x dr : L([rn - 1]) x n-1 !
L([rn - 1]) x rn-1 induces a homeomorphism Dr : |sdrL| ~=|L|.
We will be concerned in particular with the simplicial circle
S1 = Hom (-, [1])=@
7
where the identification means that maps ff : [k] ! [1] with ff(i) = 0 for
all 0 i k and maps fi : [k] ! [1] with fi(j) = 1 for all 0 j k
are identified to the basepoint of the set Hom ([k], [1]). Notice that a map
ff : [k] ! [1] is given by
(
0 0 j a - 1
ff(j) =
1 a j k
for some 0 a k + 1. Then we can identify S1([k]) with the set [[k]] via
the correspondence which sends ff to a, if 0 < a k, and sends ff to the
basepoint otherwise.
Recall that the non-degenerate simplices of S1 correspond to the maps ff
which are injective. In particular, this makes it easy to see that any simplex
of dimension two or higher must be degenerate.
Example 3.1. We want to compute the geometric realization of sd2S1. An
n-simplex of sd2S1 is an element of
Hom ([n] * [n], [1])
and there is a map i1 : [n] ! [n]*[n] (respectively, i2 : [n] ! [n]*[n]) which *
*is
the identity on the first copy of [n] (respectively, the second copy). The map
ff in Hom ([n] * [n], [1]) is non-degenerate if and only if either ff O i1 or*
* ff O i2
is injective. This makes it easy to see that all simplices of dimension two or
higher are degenerate. So to describe the geometric realization |sd2S1|, we
need only look at simplices of dimensions zero and one.
First, notice that Hom ([0] * [0], [1]) is a set of three elements, ff, fi*
* and
fl. The map ff has ff(j) = 0 for both j 2 [0] * [0] and fi has fi(j) = 1 for bo*
*th
j. These two maps are identified to the basepoint of the set sd2S1([0]). The
map fl, which sends one copy of [0] to 0 and the other to 1 is not identified
with these two since it is not a boundary element of the set. Thus there are
two 0-simplices in sd2S1.
The set Hom ([1] * [1], [1]) has five elements, and again we identify the
map which is identically 0 with the map which is identically 1. After the
identification is made, label the four remaining maps ff1, ff2, ff3 and ff4 and
let ff1 be the basepoint map. Let c0 : [1] * [1] ! [1] be the constant map
whose value is 0 and c1 : [1] * [1] ! [1] be the constant map whose value is
8
1. We can identify these:
ff1 O i1= c0 ff1 O i2= c0
ff2 O i1= c0 ff2 O i2= id[1]
ff3 O i1= c0 ff3 O i2= c1
ff4 O i1= id[1] ff4 O i2= c1
Notice that since only ff2 and ff4 are injective on one component, these must
represent the non-degenerate simplices. One can also check that ff1 and ff3 are
degenerate by noticing that they can be obtained by precomposing the maps
of Hom ([0] * [0], [1]) with the map ffi0 * ffi0 where ffi0 : [1] ! [0] is th*
*e unique
map. This specific case indicates a general method for determining the face
maps. Let oe0 : [0] ! [1] be the map whose image is 0, and oe1 : [0] ! [1] be
the map whose image is 1. The face maps are given by precomposing on the
concatenation. We have ff2O(oe0*oe0) is the basepoint 0-simplex (i.e., the map
which is constantly 0 or 1) and ff2 O (oe1 * oe1) is the non-basepoint 0-simplex
(i.e., it is the map from [0] * [0] to [1] which is both injective and surjecti*
*ve).
Similarly, ff4 O (oe0 * oe0) is the non-basepoint 0-simplex and ff4 O (oe1 * oe*
*1) is
the basepoint 0-simplex. Thus the face maps identify the end points of the
1-simplices to opposite 0-simplices. The geometric realization is a circle with
two 0-simplices and two 1-simplices, like this:
_ff2_________________________________________*
*_________________________________________________________________@
ff1 o_____________________________________________*
*_________________________________________________________________@
ff4
Let X be any other simplicial finite pointed set. Then there is a simplicial
finite pointed set, which we call the suspension, given by
S1 ^ X : ! Fin*
[n] ! S1([n]) ^ X([n]).
Corollary 3.2. |sdrS1 ^ X| ~=|S1 ^ X|.
Proof.This follows immediately from Lemma 3.1 and the fact that the smash
__
product commutes with realizations. |__|
Let L be a fixed simplicial set. In [McC ], McCarthy defines a system of
natural operators to be a family of simplicial maps OEr : sdrL ! L for each
9
r 2 Z+ , which satisfy OE1 = idL and OEr O (sdrOEs) = OErs. We want instead to
have natural operators on suspensions which satisfy the same property. In
particular, for the simplicial set L = S1^X the following diagram commutes:
sdr(ffis)^1
sdrsS1 ^ XT__________//_sdrS1 ^ X
TTT
TTTTT ffir^1||
ffirs^1TTT))TTTfflffl|
S1 ^ X
Any such family of operators gives rise to a family r of maps by letting r
be the composite
~=
|S1 ^ X| _____//_|S1| ^ |X| (2)
D-1r^1||
fflffl| |ffir|^1 ~
|sdrS1| ^ |X|_________//_|S1| ^ |X|=_//_|S1 ^ X|
Example 3.2. In this example we wish to construct a system of natural
operators OEr : sdrS1 ! S1 for which r will correspond to the r-fold covering
map of the circle. Let ik, 1 k r be the inclusion of [n] into the k-th copy
of [n] * . .*.[n] (r times). As a first step, notice that for any simplicial set
L, there is a map
sdrL ! Lxr
given by (L(i1), . .,.L(ir)). When L = S1, this map lands inside S1_. ._.S1.
To see this, note that for any ff 2 Hom ([n] * . .*.[n], [1])=@, at most one
ff O ik is surjective. The other ff O ij are identically 0 or 1, and hence these
correspond to the basepoint. Note that the vertices of sdrS1 are all sent to
the basepoint. Call this map pr : sdrS1 ! S1 _ . ._.S1.
The fold map, +, is the unique map that makes the following diagram
commute:
(S1 _ S1) (3)
u::u| ddIII
i1uuu | Ii2II
uuu | III
uu | I
S1 +| S1
JJJ | tt
JJJ | ttt
JJJ | ttt
id J%%fflffl|idzztt
S1
10
and it is associative up to homotopy. So, + : S1 _ . ._.S1 ! S1 is well
defined for arbitrary coproducts S1 _ . ._.S1. Define OEr to be the composite
+ O pr. Thus OEr : sdrS1 ! S1. One can check that
(
ff O ikif ff O ik is surjective
OEr(ff) =
* if no such k exists.
Upon applying the functor sds, the map sdsOEr is given on
ff 2 Hom ([n] * . .*.[n], [1])
(where the rs-fold concatenation of [n] with itself is in the first variable of
Hom ) by (
ff O irkif ff O irkis surjective
sdsOEr(ff) =
* if no such k exists.
where irkis the inclusion of the r-fold concatenation [n] * . .*.[n] into the
k-th block of r, for 1 k s. If there exists a k such that ff O ik : [n] ! [*
*1]
is surjective, then OEs O sdsOEr(ff) = OErs(ff) = ff O ik. If no such map exis*
*ts,
then both sdsOEr and OErs are constant maps. Since sdsOEr and OErs agree, the
collection of map OEr forms a natural system.
We then have maps r : |S1 ^ X| ! |S1 ^ X| given by the composition
in diagram (2). If X is a point, i.e., the functor which sends each [n] to {0},
then |S1 ^ X| = |S1| and this map corresponds to the r-fold covering map of
the circle. In fact, this corresponds exactly to the map which "pinches" (i.e.,
identifies) the 0-cells of |sdrS1| together and then folds together the r circl*
*es
of the resulting bouquet of circles to one circle.
We will be interested in the case X = Sd-1, in which case r is a map
from Sd to Sd. Then, r induces multiplication by r on the top dimension of
homology, since r is an r-fold covering map on the suspension coordinate.
Now we are ready to define operations on ß*(|F O L|) using the family of
operators r from Example 3.1. By Lemma 2.2, we have
a
|F |L = F (n) x |L|xn =~
n2N
In the case where L = S1 ^ X is actually a suspension, we would like to
define operations from |F |S1^X ! |F |S1^X by using the map r : |S1 ^ X| !
11
|S1 ^ X|. Abusing notation, we define r : |F |S1^X ! |F |S1^X by
a a a
r = 1x( r)xn : F ([[n]])x|S1^X|xn =~ ! F ([[n]])x|S1^X|xn =~ .
n n2N n2N
Theorem 3.3. The map r : |F |S1^X ! |F |S1^X is well defined.
Proof.To check that these operations are well defined on realizations, it
suffices to check that the operations preserve the equivalences ~ . Note that
as in Lemma 2.2, we have
a
|F |L ' F ([[n]]) x |Lxn |=~ .
n2N
Identify an element of L with the image ffi(1) for some ffi 2 HomFin* ([[1]], L*
*).
Then any collection of maps ffi, 1 i n, assemble into a map ff : [[n]]!
L. Let f : [[m]] ! [[n]]be a map in Fin*. We have f*u = F (f)(u) for
u 2 F ([[m]]). Let f*(ff1(1), . .,.ffn(1)) = ((ff O f O q1), . .,.(ff O f O qm *
*)) where
qi : [[1]]! [[m]]is the unique map with qi(1) = i. Then f*(ff1(1), . .,.ffn(1))
assembles into f*(ff) 2 HomFin* ([[m]], L). The equivalence ~ is given by
(f*u, (ff1(1), . .,.ffn(1)), t) ~ (u, f*(ff1(1), . .,.ffn(1)), t)
where t 2 k (we have expanded |Lxn | in terms of L(k)xn x k). Then,
r(f*u, (ff1(1), . .,.ffn(1)), t) =
(f*u, (OEr ^ 1) O ff1(1), . .,.(OEr ^ 1) O ffn(1), t)~
(u, (f*((OEr ^ 1) O ff1)(1), . .,.((OEr ^ 1) O ffn)(1)), t) =
(u, ((OEr ^ 1) O (ff O f O q1)(1), . .,.(OEr ^ 1) O (ff O f O qm )(1)), t)*
* =
r(u, ((ff O f O q1)(1), . .,.(ff O f O qm )(1), t) =
r(u, f*(ff1(1), . .,.ffn(1)), t)
__
Thus, r preserves the equivalence ~ as desired. |__|
Remark 3.4. The results of xx2 and 3 can be extended to functors F to
k-vector spaces. In particular, we can translate the expression
|F |L = F ([[-]]) Fin*|HomFin* ([[-]], L)|.
12
Let Func(Hom , Vk)be the category of functors from Fin* to the category of
vector spaces over k and let F 2 Func(Hom , Vk). There is a version of the
tensor product of functors in this situation analogous to the one we used for
simplicial sets which uses direct sum instead of disjoint union, tensor product
over k instead of products, and the quotient of the subspace generated by
the equivalence relation ~ rather than just the quotient by the relation ~
(see [P ] for details). Denote this tensor product by k,Fin*to distinguish it
from Fin*of x2. We have
F ([[n]]) = F ([[-]]) k,Fin*k[HomFin* ([[-]], [[n]])]
where k[HomFin* ([[-]], [[n]])] is the free abelian group on HomFin* ([[-]], [[*
*n]]).
The equivalence here is obtained by using the maps
F ([[n]])! F ([[-]]) k,Fin*k[HomFin* ([[-]], [[n]])]
u 7! u k,Fin*id[[n]]
and
F ([[-]]) k,Fin*k[HomFin* ([[-]], [[n]])]!F ([[n]])
u ff 7! ff*u
which are inverses (note the similarity with Lemma 2.1). Then we have
F O L([k]) = F ([[-]]) k,Fin*k[HomFin* ([[-]], L[k])].
Since we have previously expressed HomFin* ([[n]], L) as (L)x[[n]](see x2), we
can express F O L as a tensor product more succinctly as
|F O L| = F ([[-]]) k,Fin*k[(L[-])x[[-]]].
Here, we have written (L[-])x[[-]]to emphasize that there this is a functor
of two variables: a variable in and a variable in Fin*.
4 The chain level
We now want to show that if the operations r of the last section are extended
to this situation, they are actually well defined on the chain level. First we
13
want to define a new map r on the chain level which will agree with the
map from x3 on homology. Since r is built out of maps OEr : sdrS1 ! S1,
to show that these operations are well defined on the chain level we need an
equivalence
D(r) : C*(F O L) ! C*(F O sdrL)
to serve as the inverse of Dr : |sdrL| ! |L| in our old setting. An equivalence
such as this is provided by McCarthy [McC ]. The construction is included
here for completeness.
In x3 we discussed using the functor S1 : ! Fin* which on objects
is S1[n] = [[n]](recall that the set Hom ([n], [1]) has n + 2 elements, two
of which can be thought of as the endpoints of the unit interval and are
identified to the basepoint). Let F(M,A) be the functor from the category of
finite pointed sets to the category of A-modules which takes [[n]]to M k
A kn (i.e. there are n copies of A) where A is a commutative k-algebra and
M is an A-module. Define F(M,A)(f) for a morphism f : [[m]] ! [[n]]by
F(M,A)(a0 a1 . . .am ) = b0 b1 . . .bn where
Y
bj = ai,
i
f(i)=j
and we take bj = 1 if f-1 (j) = ;. The chain complex associated to the functor
F(M,A)O S1 is the Hochschild homology chain complex, with coefficients in
M. We want to keep this case in mind while considering the construction.
In this case, F(M,A)O S1[n] is M A n . The notation (M, A) L = F(M,A)O L
will be convenient for us to use. When M = A, we will write more simply
A L = FA O L.
1 1
We'll denote an element of the chain complex A S = FA O S by
(a0, a1, . .,.an).
Note that the element a0 corresponds to the basepoint and that the symbol
has been replaced by commas. We wish to produce a map which sends
1[n]
(a0, a1, . .,.an) into A sdrS . Note that the element (b0, b1, . .,.brn-1) of
1[n]
the chain complex A sdrS can be expressed as a rectangular r by n + 1
array. This will correspond to subdividing the edge of S1 into r pieces. The
general idea behind the construction is to count all the ways in which one
can distribute (a0, a1, . .,.an) into an r by n + 1 array, maintaining certain
14
parameters: a0 must continue to be the first entry of the array, each column
of the array can only have one entry ai (all the other entries are "1"), and
the order a0, a1, . .,.an must be preserved as one reads the array from the
upper left corner to the lower right corner. If one then expresses the array as
an r x (n + 1)-fold tensor product and sums over all possible arrays, then this
will form the chain map we seek. The formal description below of this chain
map seeks to codify this process by assigning to each element ai a column
(based on a permutation of (1, . .,.n)) and a row (based on a map from
(1, . .,.n) to [r]).
Let Hom *([n], [m]) denote the set of maps from [n] to [m] in which
send 0 to 0. Let S(r, n) be the set of pairs
(oe, f) 2 n x Hom* ([n], [r - 1])
subject to the condition that if i < j and oe(i) > oe(j), then
f(i) < f(j)
for 1 i, j n. For j = (oe, f) 2 S(r, n), define a map
fflop''2 Hom *([n], [rn + r - 1])
by
fflop''(i) = oe(i) + f(i)(n + 1)
for 1 i n. Let fflop''(0) = 0. Define a corresponding map
ffl''2 Hom *([rn + r - 1], [n])
by
ffl''(k) = max {i|fflop''(i) k}
Then we have a map Dn(r) : CnF O L ! Cn(F O sdrL) given on x 2 CnF O L
by X
Dn(r)(x) = sgn (j)ffl*''(x)
''2S(r,n)
where sgn(j) = sgn (oe) and ffl*''is the map on the chain level induced by
F O L(ffl'').
15
Suppose that L = S1. On the chain level, the term associated to
j = (oe, f)
is given by the assignment of x = (a0, . .,.an) to
(a0, b1, . .,.bn, 1, bn+1, . .,.b2n, . .,.1, b(r-1)n+1, . .,.brn)(4)
where (
ai j = oe(i) + f(i)(n + 1);
bj =
1 otherwise.
Note that this is well defined since oe(i) is not equivalent to oe(j) mod n
whenever i 6= j. Also note that for any j,
oe(i) + f(i)(n + 1) < oe(i + 1) + f(i + 1)(n + 1)
so that ffl''is order preserving.
Example 4.1. This example illustrates how j distributes an element
(a0, . .,.an) 2 Cn(FA O S1)
into an element (a0, b1, . .,.brn) (this is shorthand for Equation (4) of Cn(FA*
* O
sdrS1). First, think of (b1, . .,.brn) as an array of n columns and r rows. The
jth column has entries 1 or aff(j)such that there is only one entry in each
column not equal to 1.
Reading the array from upper left across each row to the lower right
column, the order of (a1, . .,.an) is preserved. For example, the element of
C3(sd2(S1)) given by
{ a0 1 a1 1
1 a2 1 a3 }
corresponds to the element of S(2, 3) which is the permutation (12)(3) and
the map in Hom ([3], [1]) given by sending 0 ! 0, 1 ! 0 and both 2 ! 1
and 3 ! 1 . The permutation can be read off by multiplying entries of the
columns together. The order (a1, a2, a3) has been permuted to (a2, a1, a3).
The map f 2 Hom *([3], [1]) indicates that a1 is in the first row, and a2 and
a3 are in the second row.
From [McC ] we have the following lemma:
16
Lemma 4.1. ([McC ]) The maps D*(r) assemble to give a chain map D(r)
which passes to the normalized chain complex. Furthermore, for all r 2 N,
the chain map D*(r) is a quasi-isomorphism with H*(D*(r)) = ß*(Dr)-1.
In general, for the pointed simplicial sets L = S1 ^ X (this works more
generally, but we are only concerned with this case), there is a map
D(r) ^ 1p : C*F O (S1 ^ X[p]) ! C*F O (sdrS1 ^ X[p]).
Here we are considering S1^X as a bisimplicial set and fixing the X direction,
hence C*(F O(S1^X[p])) represents a chain complex with differentials coming
from the S1 direction. Since D(r) is functorially constructed, we can then
extend this to the map
Tot(D(r) ^ 1) : Tot(C**F O S1 ^ X) ! Tot(C**F O (sdrS1 ^ X)).
The identity maps 1p assemble to give the identity map on the chain complex
in the X direction. Since D.(r) ^ 1 is a quasi-isomorphism, this map gives
a chain equivalence. Now by the Eilenberg-Zilber Theorem (see, for exam-
ple, [M ] p. 238), we have equivalences between Tot and the diagonal D for
bisimplicial sets and so we may extend D(r) ^ 1 by
TotC**F OO(S1O^ X) ___'_//TotC**F O (sdrS1 ^ X)
|'| ' ||
| fflffl|
C*F O D(S1 ^ X) _______//C*F O D(sdrS1 ^ X)
Since this is a composition of chain equivalences, the map
D(r) : C*F O (S1 ^ X) ! C*F O (sdrS1 ^ X)
is also a chain equivalence. Thus we have proved:
Lemma 4.2. There is a natural chain equivalence
D(r) ^ 1 : C*F O (S1 ^ X) ! C*F O (sdrS1 ^ X)
The ambiguity here is intentional. The map D(r) is a chain equivalence
both for Tot C**F O (S1 ^ X)) and C*F O D(S1 ^ X)). The previous two
lemmas prove:
17
Theorem 4.3. If r is given by
D(r)^1 1 F(ffir^1) 1
r : F O (S1 ^ X)__________//_F O (sdrS ^ X)________//_F O (S ^ X)
then r is well defined on the chain level.
Example 4.2. To calculate such a map we make use of the arrayed represen-
tation of D*(r) given in Example 4.1. For example, in the case where F = FA
(this functor also gives the higher order Hochschild homology chain complex
when evaluated on Sd), to calculate 2 : C*(FA O S1[3]) ! C*(FA O S1[3]), we
first use the sum of the (a0, b1, . .,.brn):
a0 a1 a2 a3 1 1 1 1
+ a0 a1 1 1 1 1 a2 a3
+ a0 a1 a2 1 1 1 1 a3
- a0 1 a1 a2 1 a3 1 1
- a0 1 a1 1 1 a2 1 a3
+ a0 1 1 a1 1 a2 a3 1
+ a0 1 1 1 1 a1 a2 a3
+ a0 a1 1 a2 1 1 a3 1
Notice that all of the members of the n + 1-st column are 1. Then OE2 is given
by multiplying the elements of the ith column , 1 i (n=r), together with
the elements of the i + n + 1st column. In this case, multiply every fourth
element together (we subdivided into two pieces). This process will work the
same way for FA O (S1 ^ X), except that since we are only subdividing the
S1 we get the array corresponding to L[n] copies of S1[n]. For example, an
element of C2FA O (sd2S1 ^ S1) might look like:
a0 | a1 1 | a3 1
1 | 1 a2 | 1 a4
where the two divisions represent the first and second copy of S1, since
S1[2] = [[2]].
5 Hopf Algebra Structure
The classical decomposition of Hochschild homology (i.e., the case FA O S1)
attained independently by Loday and Gerstenhaber-Schack (see [L ], [L2 ] or
18
[G-S ]) makes use of the Eulerian idempotents associated to the ~-operations.
These ~-operations are obtained by using the Hopf algebra structure on the
Hochschild complex, whose multiplication map will be denoted by ~ and
comultiplication will be denoted by . The r-th ~-operation, ~r, is defined
to be r iterations of the convolution operator, *, applied to the identity map.
That is, ~2 = id * id = ~ O (id id) O (the convolution of the identity with
itself) and ~n is id * . .*.id (n times). In other words, ~3 = id * id * id =
~ O (id ~ O (id id id) O id ) O , which is well defined since ~ and *
* are
associative. We wish to express this Hopf algebra structure in terms of the
pinch and the fold map on the circle. We can then use this new perspective
to extend the Hopf algebra structure to the chain complex C*FA O S1 ^ X.
McCarthy showed [McC ] that in the case of regular Hochschild homology,
the operations r and ~r are the same, up to a sign (McCarthy's operations
r are the same as the ~r which appear in [L ], but they differ by a sign
from [L2 ] - we are using the lambda operations as described in [L ]). In this
section, we hope to provide a geometric motivation for this equivalence.
Definition 5.1. A differential graded Hopf algebra over A is a homological
chain complex B* (* 0) whose entries are A-modules which has chain maps
~
B* ____//_T ot(B* A B*)___//_B*
satisfying:
i. and ~ are associative up to homotopy; that is, H*((1 ) O ) =
H*(( 1) O ).
ii. Let ø be the map which flips two factors. The following diagram commutes
up to homotopy:
1 fi 1
B* B* ____//_(B* B*) (B* B*)____//(B* B*) (B* B*).
~|| |~|~
fflffl| fflffl|
B* __________________________________________//B* B*
iii. B* is unital and counital.
We have the equivalence
1 S1 (S1_S1) S1_S1
Tot C**((A S ) A (A )) = TotC**(A A A) = TotC**A .
19
Here we mean that S1 _ S1 is the bisimplicial set with
S1 _ S1([p], [q]) = S1[p] _ S1[q].
Note that since TotC**S1_S1 is equivalent to C*D(S1_S1) via the Eilenberg-
Zilber theorem, we are justified in using coproduct notation. The coproduct
of simplicial finite pointed sets X and Y is the simplicial set with (X_Y )[n] =
X[n]_Y [n]. To alleviate ambiguity, our convention will be that X_Y denotes
a bisimplicial set, and to specify the simplicial set of one variable, we will
write D(X _ Y ).
A multiplication on the Hochschild complex is a map
1 S1 S1
Tot C**(A S A A ) ! C*A .
1 S1
That is, it is a map from the total complex of C**(A S A A ), rather
1 S1
than a map from C*D(A S A A ). However, since the diagonal and the
total complex are chain equivalent, we will ignore this ambiguity for now.
We will address this in Lemma 5.3.
1_S1) S1 S1
Since T otC**A (S = T otC**(A A A ), defining a multiplica-
tion on the Hochschild complex can be accomplished by defining a multipli-
cation S1 _ S1 ! S1. Such a map is given by the fold map. Since this map
is simplicial, it will induce a chain map after we apply FA . Let ~f be the
multiplication map induced by the fold map.
To define a comultiplication, we define
1 D(S1_S1)
C*(A S ) ! C*(A ).
by using the pinch map. When S1 is a topological space this map pinches
the basepoint to its antipodal point. However, this map is not simplicial.
Recall from Example 3.2 that we can make this map simplicial by using
subdivisions. Use the map sd2S1 ! S1 _ S1 which identifies the two 0-cells
of sd2S1. This produces
1 D(2)//_ sd S1 _____// D(S1_S1)
C*(A S ) ' C*(A 2 ) C*(A )
where the equivalence here is on the chain level, and is given via the map
D(2) from x4. Let p be the map induced by the pinch map.
With the pinch and the fold map as above we can give a geometric inter-
pretation of the Hopf algebra structure on the Hochschild complex.
20
Lemma 5.2. The chain maps ~f and p induce a commutative graded Hopf
algebra structure over A on the Hochschild chain complex, up to homotopy.
That is, H*(C*FA O S1) (with trivial differential) is a commutative graded
Hopf algebra.
Proof.It suffices to check that the following diagram commutes upon taking
homology:
1_S1) _D(2)_D(2)//_ D(sd S1_sd S1)___p_/p/_ D((S1_S1)_(S1_S1))
C*(A D(S ) ' C*(A 2 2 ) C*(A )
___
~f|| ~_______ |~f_~f|
fflffl| fflffl____ fflffl|
1 ________D(2)_____// sd S1 _______p_____// D(S1_S1)
C*(A S ) ' C*(A 2 ) C*(A )
Note that sd2D(S1 _ S1) = D(sd2S1 _ sd2S1) since
sd2D(S1 _ S1)[n] = [[rn - 1]]_ [[rn - 1]]= D(sd2S1 _ sd2S1)[n].
First we want to show that the left hand side of the rectangle commutes
on homology. We use the usual identification [K ]
H*(C*FA O S1) = ß*(|FA O S1|)
where we regard FA O S1 on the right side as a simplicial set. By Lemma
4.1, H*(D*(r)) = ß*(Dr)-1. So it suffices to show that Dr behaves well with
respect to ~f. However, this follows immediately since the map 1 x dr (the
map which defines Dr levelwise) commutes with the fold maps S1 _ S1 ! S1
and sdrS1 _ sdrS1 ! sdrS1.
Now, to show that the right hand square commutes on homology means
that the diagram commutes after ß*(| |) has been applied. But |FA O S1| =
FA [[-]] k,Fin*k[(Sd[-])x[[-]]]. So the Lemma follows by the commuting of
__
the pinch and fold maps on |S1| up to homotopy. |__|
Note that FA ( ) = A ( )is coproduct preserving, so that it takes coprod-
ucts of finite pointed sets to tensor products of A-algebras. Thus the fold map
1
of S1 induces the fold map on A S , which is exactly the multiplication map
1 S1 D(S1_S1)
~f. Here we note that the coproduct C*(A S A A ) = C*(A )
1_S1
is the diagonal of the bicomplex C**A S - there is also a multiplication
map ~ defined on the total complex (see [L ] for details). We will show in the
21
next lemma that the fold map on the diagonal extends to the multiplication
1 S1
map on the total complex of C**(A S A A ). In this way, we show that
the Hopf algebra structure we have defined is the same as that of [L ].
Lemma 5.3. The Hopf algebra structure on the Hochschild complex of Lemma
5.2 agrees with the Hopf algebra structure on the normalized Hochschild com-
plex defined by Loday and Gerstenhaber-Schack. In particular, the normalized
Hochschild complex is strictly a Hopf algebra.
Proof.To prove the lemma we need to show that ~f and p agree with
the multiplication and comultiplication defined by Loday and Gerstenhaber-
Schack on the Hochschild complex. We have already noted, that the map
~f is the fold map in the category of A-algebras, at least on homology. We
1_S1
can extend this to the total complex of C**(A S ) using the following
commutative diagram:
1 S1
TotC**(A9S9 A A )
sss eeKKKK
ssss ~=E|-Z| KKKK
sss fflffl| KKK
j1sss 1 S1 KKj2K
ssss C*D(A S A A ) KKK
sss KKKK
ssss ~=|| KKKK
sss fflffl| KK
1 _______A_i1____// D(S1_S1) oo__A_i2_______ S1
C*(A S )N C*(A ) C*(A )
NNN | pppp
NNNN | pppp
NNNN ~f| pppp
= NNNNNN || pppp=p
NNN | pppp
NN''N fflffl|wwppp
1
C*(A S )
The bottom triangles commute because D(S1 _ S1) is the coproduct of S1
with itself in the category of simplicial finite pointed sets, ~f is induced by
the fold map, ik are the natural inclusions, and FA is coproduct preserving.
The top triangles also commute and the argument follows by brute force.
Recall that the Eilenberg-Zilber chain equivalence
1 S1 S1 S1
Tot C**(A S A A ) ! C*D(A A A )
22
1 S1
is given on a b 2 Cp(A S ) Cq(A ) by
X
a b 7! s!(q). .s.!(1)a sff(p). .s.ff(1)b
(ff,!)2
(p,q)-shuffles
where the si are degeneracies and (p, q)- shuffles are pairs of monic order
preserving functions oe : {1, . .,.p} ! {1, . .,.p + q} and ! : {1, . .,.q} !
{1, . .,.p + q} such that the assignment
ff(ff,!): {1, . .,.p + q} ! {oe(1), . .,.oe(p), !(1), . .,.oe(q)}
1
is a permutation. For a 2 Cp(A S ), we have j1(a) = a 1 where 1 2
1
C0(A S ). There is only one (p, 0)-shuffle, and hence (E - Z) (j1)(a) = a
sp . .s.1(1). To see that this is (A i1)(a), note that sp . .s.1(1) is a degene*
*rate
element coming from A *, where * = 0 is the basepoint of S1. Now, the
composite ~f O E-Z and the map ~ must agree since they both satisfy the
universal property of the fold map in the category of A-algebras.
To show that the comultiplication maps are the same, we must show that
the diagram
1 _______p_______//_D(S1_S1)
C*(A sd2SO)O C*A
D2|| |E-Z|
| fflffl|
1 ____________// S1 S1
C*(A S ) Tot C**(A A A )
commutes. Again, we show this by straightforward computation.
Recall ([L ]) that the comultiplication on the Hochschild chain complex is
given by:
Xn
(a0, a1, . .,.an) = (a0, a1, . .,.ai) (1, ai+1, . .,.an)
i=0
1
for (a0, . .,.an) 2 (A S )[n]. It's enough to compute (E-Z)( p)(D(2)) for
this element.
We have from x4 that for S(2, n) n x Hom* ([n], [1])
X `a , b , . . .,b '
D(2)(a0, . .,.an) = sgn(oe) 0 1 n (5)
1, bn+1, . . .,b2n
(ff,f)2S(2,n)
23
where for a given term, bi = aj if i = oe(j) + f(j)n for some 1 i n or
bi = 1 otherwise. Again recall that only one of bi and bn+i is not 1, and that
oe(1) + f(1)n < . . .< oe(n) + f(n)n. That is, the order (a0, a1, . .,.an) is
preserved. The pinch map simply takes this sum to
(a0, b1, . .,.bn) (1, bn+1, . .,.b2n). (6)
1 S1
In this direction, for a b 2 D(A S A ), the Eilenberg-Zilber chain
equivalence is given by
Xn
a b 7! d~n-ia di0b
i=1
where d~ is the iterated last face operator (the "wrap aroundö perator).
Then, applying the chain equivalence to equation (5.2), we have for each
(oe, f)
Xn
(bi+1. .b.na0, b1, . .b.i) (bn+1 . .b.n+i, bn+i+1, . .,.b2n)
i=1
Notice here that the products bi+1. .b.na0 and bn+1 . .b.n+ihave become co-
efficients. Then, for each 0 i n, there is exactly one pair (oe, f) 2 S(2, *
*n)
which produces a non-degenerate element. Specifically, that pair is oe = id
and f(k) = 0 for k i and f(k) = 1 for k > i. Any other choice will produce
degenerate elements (that is, elements with a 1 in a non-basepoint slot) and
so on normalized chain complexes, this agrees with the comultiplication map
on the Hochschild chain complex.
__
|__|
We can also interpret the chain complex associated to FA (S1 ^ X) =
1^X) (S1^X)
A (S as a Hopf Algebra in two ways. First, we endow C**(A )
with a Hopf algebra structure up to homotopy for any simplicial finite set
X again using the pinch and fold maps of the circle. On realizations a
suspension is a co-H-space, and hence we have a map |S1 ^ X| ! |(S1 ^ X) _
(S1 ^ X)| induced by the pinch map on the suspension coordinate which
defines comultiplication. Since we have (S1 _ S1) ^ X ' (S1 ^ X) _ (S1 ^ X)
we can use the fold map to define multiplication. Let SX = S1 ^ X and note
24
that sd2(SX _ SX) ~=sd2(SX) _ sd2(SX). Let be induced by the pinch
map as follows:
1^X) ______1_(pinch^1)//_ (D(S1_S1)^X) D((S1^X)_(S1^X))
C**(A (sd2S ) C**(A ) = C**A
so that = 1 pinch^1. Also let ~ be similarly defined by the fold map. Then
the diagram
D(2) 2 sd (D(SX_SX)) _ D(SX_SX_SX_SX)
C**(A D(SX_SX) ) ___'_//C**(A 2 )___//_C**(A )
~ || ~|| ~_~||
fflffl| D(2) fflffl| fflffl|
C**(A SX ) ______'_____//C**(A sd2SX)____________//C**(A (SX_SX) )
commutes up to homotopy and so we will get a Hopf Algebra structure on
T otC**FA (S1 ^ X) for any such X (and the functor F ). In particular, we
have the following
Lemma 5.4. The total complex of the bi-complex associated to the bi-simplici*
*al
A-algebra FA (S1^X) has a Hopf algebra structure over A up to homotopy. In
particular, the higher Hochschild chain complex has a Hopf algebra structure
over A up to homotopy.
Again, we can make this a strict Hopf algebra by using the Eilenberg-
Zilber chain equivalence. The precise structure on the chain level is given
here.
1^X)
Theorem 5.5. The total complex of C**(A (S ) is a strict commutative
bigraded Hopf algebra over A.
Proof.Let the cardinality of the finite pointed set X[n] minus the basepoint
be denoted by |X[n]|. Note that with this convention of size, |S1[n]| = n.
As suggested by Lemma 5.4, we define the multiplication and comulti-
plication maps strictly using the structure associated to the S1 coordinate.
As in Lemma 5.3, the comultiplication will be defined by the commuting
diagram
1^X __________//_ (D(S1_S1))^X
TotC**(A sd2SOO ) Tot C**((A )
1 (D2^1)|| 1|(E-Z)|
| fflffl|
1^X _____________// ((S1_S1)^X
TotC**(A S ) ffi Tot C**(A )
25
Let elements of the set S1[p] ^ X[q] be denoted [[p]]x X[q] where we mean
that this set has the lexicographic order coming from the product. Let
1[p]^X[q])
(a0, a1, . .,.an) be a corresponding element of C**(A (S ), so that
n = |S1[p]| . |X[q]| = p . |X[q]|, then the comultiplication is given by:
ffi(a0, a1, . .,.an) =
X
(a0, a1, . .a.i, ap+1, . .,.ap+i, . .,.a(|X[s]|-1)(p)+1, . .,.a(|X[s]|-*
*1)(p)+i)
0 i p
(1, ai+1, . .,.ap, ap+i+1, . .,.a2p, . .,.a(|X[s]|-1)(p)+i+1, . .,.a|X*
*[s]|p).
It is helpful to check that when X is trivial, this gives the comultiplication
on the Hochschild chain complex.
Similarly, the multiplication m is defined by the diagram
1_S1)^X ____m_____//_ S1^X
TotC***(A (S ) 4C**(A4 )
iiiii
iii
1 (E-Z^1)|| iiiii~i
fflffl|iii
1^S1))^X
C**(A (D(S )
Again, it is easier to clarify this diagram by defining m on elements
1[p]^X[q])
(a0, a1, . .,.an) 2 C**(A (S )
and 1 0 0
(b0, b1, . .,.bn0) 2 A (S [p ]^X[q.])
For these, ~ is given by
~((a1, . .,.an) (b1, . .,.bn0)) = 0
whenever q 6= q0. One can see that this must be the case since the Eilenberg-
Zilber equivalence we wish to use holds the X direction fixed. If q = q0, then
the multiplication is given by:
~((a1, . .,.an) (b1, . .,.bn0)) =
X
sgn(ff(ff,!))(a0b0, aff(1), . .,.aff(r), b!(1), . .,.b!(r0), . .,.
(ff,!)2
(p, p0)-shuffles
. . ., aff((|X[s]|-1)r+1), . .,.aff(|X[s]|r), b!((|X[s]-1)r0+1), . .,*
*.b!(|X[s]|r0))
26
Again, one can check that this gives the correct multiplication on Hochschild
homology by setting X = *.
1^X)
Now to show that this defines a Hopf algebra structure on T otC**(A (S ),
we need to show that the multiplication map is a coalgebra map, or equiva-
lently, that the comultiplication map is an algebra map. However, since the
multiplication and comultiplication maps have been defined on the S1 coordi-
nate alone, this follows immediately from the fact that Hochschild homology
has a Hopf algebra structure.
__
|__|
6 Eulerian idempotent decomposition
In a commutative Hopf algebra H, there is an operation on the self maps of
H called the convolution. Let ~ be the multiplication map on H and be
the comultiplication. The convolution of two maps f and g is
f * g = ~ O (f g) O .
The convolution operator is an associative operation, so that f * g * h is
well defined. When H is C*FA O S1, the r-th ~-operation is defined to be
~r = id*r (this differs from the original_definition of ~r of [G-S ] and [L2 ] *
*but
r r r 1 1
it is equivalent, see [L ]). Let ~ = C*FA_(OE ) O D*(r), where OE : sdrS ! S
r 1
is the map of Example 3.2. Notice that_~ is a self map of C*FA O S . From
r r
Example 3.8 of [McC ] we know that ~ = ~ (up to a sign). We can extend
the ~-operations to C*FA O (S1 ^ X) by setting ~r = C*FA (OEr ^ 1).
1^X
Because ~r is defined on the chain level, it induces a self map of HHS* .
1^X
Since HHS* (A) is a commutative graded Hopf algebra, there exist Eulerian
idempotents e(i)with the relationship for n 1
~r|HHS1^Xn(A) = re(1)n+ . .+.rne(n)n (7)
where e(i)n= e(i)|HHS1^Xn(A)[L ].
1^X
Theorem 6.1. Let X be any simplicial finite pointed set. Then HHS* (A)
has a decomposition as
1^X (1) S1^X (n) S1^X
HHSn (A) = en HHn (A) . . .en HHn (A)
27
1^X (r) 1
where e(r)nHHSn (A) is the n-th homology of the chain complex en FA O(S ^
X).
Proof.First observe that by Loday ([L ] Proposition 4.5.9), the idempotents
commute with the boundary of the Hochschild complex, which is induced by
the face maps of S1. Then since ^ is a bifunctor, the idempotents will also
commute with the boundary of the higher Hochschild chain complex. We
have
e(1)n+ . .+.e(n)n= id,
so that the Eulerian idempotents split FA O (S1 ^ X)[n] into
e(1)nFA O (S1 ^ X)[n] . . .e(n)nFA O (S1 ^ X)[n]
__
and taking homology, this yields the desired decomposition. |__|
The following Corollary is the special case X = Sd-1.
Corollary 6.2. We have a decomposition of higher Hochschild homology
given by
d (1) Sd (n) Sd
HHSn(A, M) = en HHn (A, M) . . .en HHn (A, M) .
Observe that because of the relationship (1) given above, we have that
d
~r acts by multiplication by rk on e(k)nHHSn(A, M).
7 Agreement of the Decompositions
The goal of this section is to show that the operators r act as "eigenweights"
d
on two decompositions of HHS . The first decomposition is obtained by con-
sidering the Eulerian idempotents in x5 and the second decomposition is given
by Pirashvili in [P ]. One important difference between the two decomposi-
tions is that by using the Eulerian idempotents, one finds decompositions of
1^X
HHS for any simplicial set X. However, the decomposition obtained by
d
Pirashvili applies only to spheres, HHS . Therefore in this section, we will
restrict our attention to the functors FA O Sd. We will show that when we
set X = Sd-1 in the first decompositions, the first decomposition agrees with
the second decomposition. We will recall Pirashvili's decomposition here.
28
Let B* be the chain complex in Func(Hom , Vk) defined by Bn(m) =
k[(Sd[m])[[n]]]. Pirashvili's spectral sequence is the hyperhomology spectral
sequence obtained from FA k,Fin*B* by taking a projective resolution, (FA )*,
of FA in the category of functors from finite pointed sets to vector spaces (no*
*te
that there are enough projective generators by Yoneda's Lemma - since vector
spaces have enough projective generators, the category of functors does too).
This spectral sequence exists for any simplicial set L, but in the case of
spheres, Pirashvili shows that the spectral sequence collapses at the E2 page:
Proposition 7.1 (Proposition 1.6 of [P ]). There is a first quadrant spec-
tral sequence with
E2p,q= TorFin*p(Hq(B*), FA )
converging to Hp+q(B* k,Fin*FA ). This spectral sequence collapses at the E2
page and one has the decomposition
Hn(B* k,Fin*FA ) ' p+dj=nTorFin*p(Hdj(B*), F ).
The bi-functor TorFin*prefers to the derived functors of the tensor product
k,Fin*. See [P ] for details.
We want to consider the image of r on this decomposition. To do so,
we must first understand r(H*(B*)). Then, since we have shown that r is
defined on the chain level, and since r acts on F k,Fin*B* by 1 k,Fin*OEr,
we will be able to understand the behavior of r on TorFin*p(Hdj(B*), FA ).
Lemma 7.2. The map r acts on (Tor Fin*p(Hdj(B*), FA ) by multiplication
by rj.
Proof.For any n, H*(B*)[[n]]= H*((Sd)x[[n]]). Using the Künneth theorem,
we have H*((Sd)x[[n]]) = H*(Sd) n . We'll write H*(Sd) = k kx where x is
a generator of dimension d. Then
H*(Sd) n = nj=0 ((n kx . . .kx
j))
where we mean kx . . .kx to be the j-fold tensor product. Consequently,
H*((Sd)x[[n]]) = 0 if * 6= dj for some j and Hdj((Sd)x[[n]]) = ((n kx . . .kx.
j))
Now, r acts on H*(Sd) by multiplication by r on the top dimension,
since r is the r-fold covering map on the first coordinate of S1 ^ Sd-1.
Then, writing H*(Sd) = k kx, we have that r acts as the identity on
29
k and by multiplication by r on kx. Then we can see that r applied to
Hdj((Sd)x[[n]]) acts by multiplication by rj. Note that this action does not
depend on n, so that actually we see that r acts by multiplication by rj on
(Hdj(B*)).
Now, since r is defined on the chain level, this computation passes to
__
the Tor groups so that we obtain the result. |__|
Now we can use linear algebra to conclude the
Theorem 7.3. The two decompositions of higher order Hochschild homology
agree. That is, d
e(j)nHHSn(A) = TorFin*p(Hdj(B*), FA ).
Proof.The map ~r = r plays the role of a linear operator on the k-vector
d r
space HHSn(A). Since k is a field, the image of determines its subspace
__
decomposition. The result follows. |__|
Notice that this means that
d
e(j)nHHSn
vanishes whenever j > (d=n). This is a mysterious phenomenon. Joint work
with R. McCarthy indicates that an explanation of this may be given by
looking at the decomposition of higher order Hochschild homology in another
way (using Goodwillie calculus).
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