Deformation Retracts and the Hochschild Homology
of Polynomial Rings
Ayelet Lindenstrauss*
University of Pennsylvania
Philadelphia, PA 19104
x0. Introduction
In computations of Hochschild homology and its variants, such as cyclic hom*
*ology,
neither the standard Hochschild complex
. .-.d3!A A A-d2!A A-d1!A-! 0
nor the reduced Hochschild (bar) complex
(0:1) . .-.d3!A A A-d2!A A-d1!A-! 0
is very often used. Typically, a much smaller direct summand of the complex, w*
*hich is
quasi-isomorphic to the whole, is used instead. For example, if A is a smooth n*
*-dimensional
commutative algebra over a commutative Q-algebra k, the complex with zero diffe*
*rentials
0 ! nA=k-0!n-1A=k0-!.-.0.!1A=k-0!A ! 0
can be realized as a subcomplex of (0.1) via i : rA=k! A A(r) sending
X
f0 df1 ^ . .^.dfr 7! 1_r! sgn(oe)f0 foe(1) . . .foe(r)
oe2Sr
and as a quotient of (0.1) via ss : A A(r) ! rA=ksending
f0 f1 . . .fk 7! f0 df1 ^ . .^.dfr :
Clearly ssOi = idrA=k, and by the Hochschild-Kostant-Rosenberg theorem, i and s*
*s are
actually quasi-isomorphisms. Following Kassel [1], we define a deformation retr*
*act (of the
reduced Hochschild complex) to be a complex (X:; dX ) together with chain maps *
*i:: X:!
A A(.) and ss:: A A(.) ! X: and a chain homotopy K:: A A(.) ! A A(.+1)
such that
ssrOir = idXr ; dr+1OKr + Kr-1Odr = idAA(r) - irOssr :
_________________________
* Partially supported by N.S.F. Grant No. DMS 92-03398
1
Hitherto, there have been ad hoc computations of deformation retracts, but *
*no attempt
to systematize the construction. This paper is a step in the latter direction. *
*Given a k-
algebra A with unit and a deformation retract for A, we construct wuch retracts*
* for A[x]
and A[t; t-1]. The latter construction can be adapted to give, for each Banach *
*algebra A
and deformation retract of the continuous reduced Hochschild complex for A, a d*
*eformation
retract for the ring C1 [S1; A] of smooth A-valued functions on the circle. It *
*seems likely
that similar methods will work for the non-commutative polynomial rings
A{x}=(ax - xOE(a)) ;
where OE is an automorphism of A. A more difficult question, which we do not di*
*scuss, is
the construction of deformation retracts for A=a . A, where a is an element of *
*the center of
A. It would be nice to have a general construction that specializes to Wolffhar*
*dt's complex
for coordinate rings of local complete intersections.
Finally, it should be noted that the results of this paper do not advance t*
*he state of
the art in computing Hochschild homology since the K"unneth formula already exp*
*resses
HH *(A[x]) and HH *(A[t; t-1]) in terms of HH *(A). The results which follow do*
*, however,
provide information about explicit homotopies that the K"unneth formula does no*
*t. Already
in the case of k[x; y], a homotopy contracting the reduced Hochschild bar compl*
*ex to the
Hochschild-Kostant-Rosenberg complex is quite difficult to find. The methods of*
* this paper
give a general inductive construction.
I would like to thank Michael Larsen for several useful conversations durin*
*g the com-
position of this paper.
x1. Going from a ring A to A[x]
In this section, we will assume given a commutative ground ring k with a un*
*it,a k-
algebra A, also with unit (so that if we multiply that unit by any element of k*
* we get an
element in the center of A), and a complex (X: ; dX ) whose homology is HH *(A)*
* (over
k). Often such a complex arises by tensoring a projective resolution of A, as a*
*n A Aop-
module, over A Aop with A, but this will not be assumed in the calculations an*
*d will be
used only to motivate the construtctions. What we do want is that the complex X*
*: should
be a deformation retract of the usual reduced bar complex calculating HH *(A),
(1:1) . .d.3-!A A(2) -d2!A A-d1!A-! 0
with A = A=(k . 1A ), and
dr : A A(r) ! A A(r-1)
is given by
Xr
dr = (-1)id(i) ;
i=0
d(i)(a0 a1 . . .ar) = a0 a1 . . .aiai+1 ai+2 . . .ar ; 0 i r - 1
2
d(r)(a0 a1 . . .ar) = ara0 a1 . . .ar-1 :
In some of the calculations, it will also be useful to define the operator b0: *
*A A(r) !
A A(r-1) (or sometimes b0: A(r+1) ! A(r) ) given by
r-1X
(1:2) b0= (-1)id(i) :
i=0
As a deformation retract, X: comes with chain maps iold:, ssold:,
(1:3) ioldr: Xr ! A A(r)
(1:4) ssoldr: A A(r) ! Xr
such that
ssoldrOioldr= idXr
and
ioldrOssoldr' idAA(r)
for every r 0, and a homotopy Kold:consisting of maps Koldr: A A(r) ! A A(r+*
*1)
such that
(1:5) dr+1OKoldr+ Koldr-1Odr = idAA(r) - ioldrOssoldr :
We will use this data to obtain a deformation retract of the reduced bar co*
*mplex for
the ring A[x], which would again be smaller and easier to calculate with than t*
*he whole
complex.
To do this, we start with the following free k[x] k[x]-resolution of k[x]:
.(1x-x1) mult
(1:6) 0-! k[x] k[x]-------! k[x] k[x]---! k[x]-! 0 :
After tensoring over k[x] k[x] with k[x], we obtain the complex
0-! k[x]-0! k[x]-! 0
If we had a small resolution of A which gave us X: in the tensored form, we can*
* tensor
it with the small resolution of k[x] in (1.6) to obtain a resolution of A[x]. *
* Tensoring
the resulting complex with A[x], we obtain the following complex whose homology*
* is
HH *(A[x]):
(1:7) . .-.d!X3[x] X2[x]dx- d!X2[x] X1[x]dx- d!X1[x] X0[x]dx- d!X0[x] ! 0
3
where for all ff 2 X: and i 0,
d(ffxi) = dX (ff)xi ; d(ffxidx) = dX (ff)xidx :
dx in this context can be looked upon as a place marker of degree 1. Note that*
* the
homology of the complex (1.7) gives the Hochschild homology of A[x] by constru*
*ction in
the case where X:comes from a projective A[x] Aop[x] resolution of A[x], but a*
*lso works
for any complex X: which is a deformation retract of the reduced bar complex be*
*cause of
the calculations which follow.
For the new complex (1.7), we define maps ____
inewr: Xr[x] Xr-1[x]dx ! A[x] A[x](r)
given by
inewr(ffrxi) = xiioldr(ffr) 8ffr 2 Xr; i 0
(1:8) inewr(ffr-1xidx) = xiioldr(ffr-1) * (1 x) 8ffr-1 2 Xr-1; i 0 ;
where * denotes the shuffle product. We also define
____(r)
ssnewr: A[x] A[x] ! Xr[x] Xr-1[x]dx
by
ssnewr(a0xn0a1xn1 . . .arxnr)
(1:9) = xn0+n1+...+nrssoldr(a0 a1 . . .ar)
+ nrarxn0+n1+...+nr-1ssoldr-1(a0 a1 . . .ar-1)dx *
* :
Direct calculation shows that inew:, ssnew:are chain maps. Clearly,
(1:10) ssnewrOinewr(ffrxi) = ssnewr(xiioldr(ffr)) = xissoldrOioldr(ffr) =*
* ffrxi
and
ssnewrOinewr(ffr-1xidx) = ssnewr(xiioldr(ffr-1) * (1 x))
(1:11)
= xi+1ssoldr(ioldr(ffr-1) * (1 1)) + xissoldrOioldr(ffr-1)dx*
* = ffr-1xidx
(because we are using the reduced complex) so
(1:12) ssnewrOinewr= idXr[x]Xr-1[x]dx :
*
* ____(r)
We define an explicit homotopy Knew:between inewrOssnewrand the identity of A[x*
*]A[x] ,
using the homotopy Kold:which we had for the ring A, by
K newr(a0xn0 a1xn1 . . .arxnr)
Xr nj-1X
= (-1)j xn0+n1+...+nj-1+i[(a0 a1 . . .aj-1)* (1 x)]
(1:13) j=1 i=0
ajxnj-1-i aj+1xnj+1 . . .arxnr
+ nrarxn0+n1+...+nr-1Koldl-1(a0 a1 . . .ar-1) * (1 x)
+ xn0+n1+...+nrKoldr(a0 a1 . . .ar)
for all r 0, a0; a1; : :;:ar 2 A, n0; n1; : :;:nr 0.
4
Claim: dr+1OKnewr+ Knewr-1Odr = idA[x]____A[x](r)- inewrOssnewr :
Proof. Set
F (a0xn0 a1xn1 . . .arxnr)
new new new new n n n
= dr+1OKr + Kr-1Odr - id+ ir Ossr (a0x 0 a1x 1 . . .arx r) :
We will show that F 0. The proof will proceed by induction on r; for each *
*r the
proof will be by induction on nr; nr-1; : :;:n1; n0, in that order.
The Case of r = 0:
In this case we must check
d1OKnew0(a0xn0) ?=a0xn0 - xn0ssold0Oiold0(a0)
which holds since Kold0= id- ssold0Oiold0.
The Inductive Step, r > 0:
If nr = nr-1 = . .=.n1 = n0 = 0, we have
old old old old
F (a0 a1 . . .ar) = dr+1OKr + Kr-1Odr - id+ ir Ossr (a0 a1 . . .ar) = 0
by the choice of Kold:. We prove the general case by induction on the ni, in th*
*ree steps:
First Step: If nr > 0 and we know that
F (a0 a1 . . .arxnr-1) = 0
then
F (a0 a1 . . .arxnr) = 0 :
Proof. Since all the maps d:, Kold:, iold:, ssold:commute with multiplication *
*of the first
coordinate in each monomial by x,
F (a0x a1 . . .arxnr-1) = x . F (a0 a1 . . .arxnr-1) = 0
by the inductive hypothesis.
Therefore, proving that F (a0 a1 . . .arxnr) = 0 is equivalent to proving*
* that
F (a0 a1 . . .arxnr - a0x a1 . . .arxnr-1) = 0 :
We set
= a0 a1 . . .arxnr - a0x a1 . . .arxnr-1 :
Then
Kr() = (-1)r [(a0 a1 . . .ar-1) * (1 x)] arxnr-1
+ arxnr-1Koldr-1(a0 a1 . . .ar-1) * (1 x) ;
5
so
dr+1OKr() = (-1)r[b0(a0 a1 . . .ar-1) * (1 x)] arxnr-1
- a0x a1 . . .ar-1 arxnr-1
(1:14) - [(a0 a1 . . .ar-2) * (1 x)] ar-1arxnr-1
+ a0 a1 . . .ar-1 arxnr - (arxnr-1a0 a1 . . .ar-1) * (1 x)
+ arxnr-1drOKoldr-1(a0 a1 . . .ar-1) * (1 x) :
Here b0is the operator defined in (1.2); we are using the fact that the Hochsch*
*ild boundary
map d: satisfies the Leibnitz rule with respect to the shuffle product *, and t*
*he fact that
d1(1 x) = 0.
Now by a similar calculation,
Kr-1O dr() = (-1)r-1 [b0(a0 a1 . . .ar-1) * (1 x)] arxnr-1
+ a0x a1 . . .ar-1 arxnr-1
(1:15) - [(a0 a1 . . .ar-2) * (1 x)] ar-1arxnr-1
- arKr-1(0) + arxnr-1Koldr-2(b0(a0 a1 . . .ar-1)) * (1 x)
+ (-1)r-1arxnr-1Koldr-2(ar-1a0 a1 . . .ar-2) * (1 x) :
So
(dr+1OKr + Kr-1Odr)() = -a0x a1 . . .ar-1 arxnr-1
+ a0 a1 . . .ar-1 arxnr + arxnr-1drOKoldr-1(a0 a1 . . .ar-1)
+ arxnr-1Koldr-2(b0(a0 a1 . . .ar-1)) * (1 x)
(1:16) + (-1)r-1arxnr-1Koldr-2(ar-1a0 a1 . . .ar-2) * (1 x)
= -a0x a1 . . .ar-1 arxnr-1 + a0 a1 . . .ar-1 arxnr
old old
- arxnr-1 (id- drOKr-1 - Kr-2Odr-1)(a0 a1 . . .ar-1) * (1 x)
old old
= - arxnr-1 ir-1Ossr-1(a0 a1 . . .ar-1)* (1 x)
by the inductive hypothesis. But since
inewrOssnewr()i=newr(arxnr-1ssoldr-1(a0 a1 . . .ar-1)dx)
old old
= arxnr-1 ir-1Ossr-1(a0 a1 . . .ar-1)* (1 x) ;
this calculation shows exactly that
old old old old
F () = dr+1OKr + Kr-1Odr - id+ ir Ossr () = 0 :
Second Step: If 1 j r - 1 and nj > 0, and we know that
F (a0 a1 . . .aj-1 ajxnj-1 aj+1xnj+1 . . .arxnr) = 0
6
then
F (a0 a1 . . .aj-1 ajxnj aj+1xnj+1 . . .arxnr) = 0 :
Proof. As in the first step, we have
F (a0x a1 . . .aj-1 ajxnj-1 aj+1xnj+1 . . .arxnr)
= x . F (a0 a1 . . .aj-1 ajxnj-1 aj+1xnj+1 . . .arxnr) = 0
so we set
= a0 a1 . . .aj-1 ajxnj aj+1xnj+1 . . .arxnr
- a0x a1 . . .aj-1 ajxnj-1 aj+1xnj+1 . . .arxnr
and prove that F () = 0.
Knewr() = (-1)j[(a0 a1 . . .aj-1) * (1 x)] ajxnj-1 aj+1xnj+1 . . .arxnr
and therefore we get that
(1:17)
dr+1OKnewr()
h
= (-1)j [b0(a0 . . .aj-1) * (1 x)] ajxnj-1 aj+1xnj+1 . . .arxnr
+ (-1)j-1a0x a1 . . .aj-1 ajxnj-1 aj+1xnj+1 . . .arxnr
+ (-1)j+1 [(a0 . . .aj-2) * (1 x)] aj-1ajxnj-1 aj+1xnj+1 . . .arxnr
+ (-1)ja0 . . .aj-1 ajxnj aj+1xnj+1 . . .arxnr
+ (-1)j[(a0 . . .aj-1) * (1 x)] b0(ajxnj-1 aj+1xnj+1 . . .arxnr)
i
+ (-1)r+1arxnr [(a0 . . .aj-1) * (1 x)] ajxnj-1 . . .ar-1xnr-1
and that
(1:18)
Knewr-1Odr()
= (-1)j-1 [b0(a0 . . .aj-1) * (1 x)] ajxnj-1 aj+1xnj+1 . . .arxnr
+ [(a0 . . .aj-2) * (1 x)] aj-1ajxnj-1 aj+1xnj+1 . . .arxnr
- [(a0 . . .aj-1) * (1 x)] b0(ajxnj-1 aj+1xnj+1 . . .arxnr)
+ (-1)j+rarxnr [(a0 . . .aj-1) * (1 x)] ajxnj-1 . . .ar-1xnr-1 :
Adding the last two equations, we get
(dr+1OKnewr+ Knewr-1Odr)() = :
Inspection of the formulae for inewrand ssnewrshows that
inewrOssnewr() = 0
7
and thus we obtain the desired equality F () = 0.
Third Step: For any n0 0,
F (a0xn0 a1xn1 . . .arxnr) = 0 :
Proof.
F (a0xn0 a1xn1 . . .arxnr) = xn0F (a0 a1xn1 . . .arxnr) = 0
by the previous steps. *
* tu
x2. Going from a ring A to A[t; t-1]
In this section we will assume given, as in the preceding section, a k-alge*
*bra A with
a deformation retract X: of the reduced bar complex of A with the maps iold:, s*
*sold:, and
Kold:(as in (1.3), (1.4), and (1.5)).
We use these to construct a complex calculating HH *(A[t; t-1]) which is a *
*direct
summand in the reduced bar complex, with the corresponding inclusion and projec*
*tion
maps, and a homotopy Knew:which will show that the direct summand is indeed qua*
*si-
isomorphic to the entire reduced bar complex.
In this case, again we use for motivation a free k[t; t-1]k[t; t-1]-resolut*
*ion of k[t; t-1]:
.(1t-t1) mult
(2:1) 0-! k[t; t-1] k[t; t-1]-------! k[t; t-1] k[t; t-1]---! k[t; t-1]-*
*! 0 :
which we tensor over k[t; t-1] k[t; t-1] with k[t; t-1] to obtain the complex
0-! k[t; t-1]-0! k[t; t-1]-! 0 :
We tensor the small A Aop-projective resolution of A which gave us X: in the t*
*ensored
form (if such a resolution existed) with the small resolution of k[t; t-1] in *
*(2.1) to obtain
an A[t; t-1] Aop[t; t-1] resolution of A[t; t-1], and tensoring the resulting *
*complex with
A[t; t-1], we obtain the following complex whose homology is HH *(A[t; t-1]):
(2:2) . .-.d!X2[t; t-1] X1[t; t-1]dt-d!X1[t; t-1] X0[t; t-1]dt-d!X0[t; t-1*
*] ! 0
where for all ff 2 X: and i 0,
d(ffti) = dX (ff)ti ; d(fftidt) = dX (ff)tidt :
We define chain maps inew:and ssnew:as in (1.8) and (1.9), and observe th*
*at the
calculations of (1.10) and (1.11) continue to hold as formal equalities even *
*when when
we allow the variable t to have negative exponents; we therefore know that, as *
*before, the
relation ssnewrOinewr= idXr[t;t-1]Xr-1[t;t-1]dtholds for all r 0.
8
It should be noted that as part of the definition of ssnewr, we include a t*
*erm of the form
nrtn0+n1+...+nr-1; we could equally have taken -nrtn0+n1+...+nr+1, but in makin*
*g either
choice the symmetry between t and t-1 is broken.
To show that inewrOssnewr' id, we let
K newr(a0tn0 a1tn1 . . .artnr)
Xr ( P nj-1 if n 0)
= (-1)j P i=0-1 j tn0+n1+...+nj-1+i
if nj 0
(2:3) j=1 i=nj
[(a0 a1 . . .aj-1) * (1 t)] ajtnj-1-i aj+1tnj+1 . . .artnr
+ nrartn0+n1+...+nr-1Koldl-1(a0 a1 . . .ar-1) * (1 t)
+ tn0+n1+...+nrKoldr(a0 a1 . . .ar)
for all r 0, a0; a1; : :;:ar 2 A, n0; n1; : :;:nr 2 Z.
Claim: dr+1OKnewr+ Knewr-1Odr = idA[t;t-1]______A[t;t-1](r)- inewrOssnewr :
Proof. We set, as before,
F (a0tn0 a1tn1 . . .artnr)
new new new new n n n
= dr+1OKr + Kr-1Odr - id+ ir Ossr (a0t 0 a1t 1 . . .art r)
and show that F 0, by induction on r, and for each r by induction on |nr|; |nr*
*-1|; : :;:|n1|,
and |n0|.
The Case of r = 0:
Follows, as before, from the properties of Kold:.
The Inductive Step, r > 0:
If nr = nr-1 = . .=.n1 = n0 = 0, we have
old old old old
F (a0 a1 . . .ar) = dr+1OKr + Kr-1Odr - id+ ir Ossr (a0 a1 . . .ar) = 0
by the choice of Kold:. The general case is proved by induction on the absolute*
* values |ni|,
in three steps:
First Step: If nr > 0 and we know that
F (a0 a1 . . .artnr-1) = 0 ;
or nr < 0 and we know that
F (a0 a1 . . .artnr+1) = 0 ;
then
F (a0 a1 . . .artnr) = 0 :
9
Proof. If nr > 0, the statement is exactly that of the first step in the proof*
* of the claim
in the previous section; therefore we need only prove the result in the case nr*
* < 0.
We set
(2:4) = a0 a1 . . .ar-1 artnr+1 - a0t a1 . . .ar-1 artnr
and note that
Kr() = (-1)r [(a0 a1 . . .ar-1) * (1 t)] artnr-1
(2:5)
+ artnr-1Koldr-1(a0 a1 . . .ar-1 * (1 t) :
Similar formulae hold for KrOd(i), 0 i < r (and d(r)() = 0), so the exact argu*
*ment in
formulae (1.14), (1.15), and (1.16) carries through and gives us F () = 0. W*
*e then use
the fact that F (a0 a1 . . .artnr+1) = 0 to obtain
F (a0 a1 . . .ar-1 artnr = t-1F (a0t a1 . . .ar-1 artnr) = 0 :
Second Step: For 1 j r - 1, if nj > 0 and we know that
F (a0 a1 . . .aj-1 ajtnj-1 aj+1tnj+1 . . .artnr) = 0 ;
or nj < 0, and we know that
F (a0 a1 . . .aj-1 ajtnj+1 aj+1tnj+1 . . .artnr) = 0 ;
then
F (a0 a1 . . .aj-1 ajtnj aj+1tnj+1 . . .artnr) = 0 :
Proof. If nj > 0, the proof of the above statement proceeds exactly along the *
*lines of the
argument in the second step in the proof of the A[x] case in the previous secti*
*on; the fact
that nj+1; nj+2; : :;:nr can, in this case, be negative does not change the arg*
*ument.
If nj < 0, the higher exponents nj+1; nj+2; : :;:nr do not change anything *
*either; the
proof is adapted from (1.17), (1.18) by defining and calculating Kr() as in (2*
*.4) and
(2.5).
Third Step: For any n0 2 Z,
F (a0tn0 a1tn1 . . .artnr) = 0 :
Proof.
F (a0tn0 a1tn1 . . .artnr) = tn0F (a0 a1tn1 . . .artnr) = 0
by the previous steps. *
* tu
10
x3. Going from a Banach algebra A to C1 [S1; A]
In this section we start with a Banach algebra A, and consider the extensio*
*n of A
which we get by taking C1 [S1; A], the space of infinitely differentiable maps *
*from the
circle to A. When equipped with the supremum norm and with pointwise multiplica*
*tion,
C1 [S1; A] becomes a Banach algebra too; A is embedded in C1 [S1; A] as the sub*
*-algebra
of all constant functions.
The Hochschild homology of such an algebra A is huge, and most of the class*
*es in it
are not boundaries because the `elements' which `should' map to them by the bou*
*ndary
map of the usual Hochschild complex are actually infinite sums of tensored mono*
*mials. To
correct this, we will continuous Hochschild homology, which is defined with the*
* topological
projective tensor product ^ ssin place of the usual (see [2], section 5.6.2). *
* ^sshas the
property that
(3:1) C1 (V ) ^ssC1 (V ) ~=C1 (V x V )
for any compact smooth manifold V (where C1 (V ) refers to the ring of smooth c*
*omplex-
valued functions). A partition of unity argument shows that for any such V an*
*d any
Banach algebra A,
(3:2) C1 (V ; A) ~=C1 (V ) ^ssA :
From (3.1) and (3.2), we obtain
C1 [S1; A]^ ss(r+1)~=C1 (Tr+1; A_^ssA_^ss._.^.ssA-z_______") ;
r+1 times
where Tr+1 is the (r + 1)-torus. We define
si: C1 (Tr; A_^ssA_^ss._.^.ssA-z_______") ! C1 (Tr+1; A_^ssA_^ss._.^*
*.ssA-z_______")
r times r+1 times
for 0 1 r - 1 by letting
(si(f0 f1 . . .fr-1))(t0; t1; : :;:tr)
= f0(t0) f1(t1) . . .fi(ti) 1 fi+1(ti+2) . . .fr-1(tr) *
* :
This gives us the formula
__________^
C1 [S1; A] ^ssC1 [S1; A]ss(r)~=C1 (Tr+1; A ^ssA ^ss. .^.ssA)= [r-1i=0Im(s*
*i)
for the modules appearing in the reduced bar complex; note that Im(si) is close*
*d for every
i as the image of a complete space, and so the linear span of the finite union *
*[r-1i=0Im(si)
is also closed.
11
P r
The boundary map for this reduced bar complex is as usual dr = i=0(-1)id(*
*i),
where for 0 i < r,
(d(i)(f0 . . .fr))(t0; t1; : :;:tr-1)
= f0(t0) f1(t1) . . .fi(ti)fi+1(ti) fi+2(ti+1) . . .fr(tr-1)
and
(d(r)(f0 . . .fr))(t0; t1; : :;:tr-1) = fr(t0)f0(t0) f1(t1) . . .. . .fr-1*
*(tr-1) :
The argument which shows that the reduced bar complex has the same homology*
* as
the standard bar complex when we use ^ ssis identical to the argument when we u*
*se :
the kernels of the reduction map form an acyclic complex.
We assume, as usual, that we have a deformation retract X:of the reduced ba*
*r complex
of A with the inclusion and projection maps iold:and ssold:and homotopy Kold:.
We construct a new complex whose homology will be proven to be HH *(C1 [S1;*
* A]),
the complex
(3:3)
. .-.d!C1 (S1; X2) C1 (S1; X1)dt-d!C1 (S1; X1) C1 (S1; X0)dt-d!C1 (S1; X0) !*
* 0
where for all f : S1 ! X: and t 2 S1
(d(f))(t) = dX (f(t)) ; (d(fdt))(t) = dX (f(t))dt :
inew:and ssnew:are also defined analogously to what we have done before- fo*
*r f : S1 !
Xr and (t0; t1; : :;:tr) 2 Tr+1,
(inewr(f))(t0; t1; : :;:tr) = ioldr(f(t0))
P ` (a) (a) (a)
and if we write ioldr(f(t0)) = a=1b0 b1 . . .br ,
(inewr(fdt))(t0; t1; : :;:tr; tr+1)
Xr X`
= (-1)r-i b(a)0 b(a)1 . . .b(a)i ti+1 b(a)i+1 . . .b(a)r
i=0 a=1
where each ti+1 is regarded as an element of A via the inclusion of S1 in C as *
*the unit
circle and the inclusion of C in A as multiples of the unit element 1A .
For a monomial f0 f1 . . .fr : Tr+1 ! A ^ssA ^ss. .^.ssA, we let
(ssnewr(f0 f1 . . .fr))(t)
= ssoldr(f0(t) f1(t) . . .fr(t)) + dfr_dt(t)ssoldr-1(f0(t) f1(t) . *
*. .fr-1(t))dt :
12
The desired relation
ssnewrOinewr= idC1 (S1;Xr)C1 (S1;Xr-1)dt
still holds, by calculations exactly analogous to (1.10) and (1.11), but seeing*
* this requires
an extra step. The point is that in our definition of ssnewrabove, we are looki*
*ng at monomials
that take (t0; t1; : :;:tr) to f0(t0) f1(t1) . . .fr(tr), and the form in whi*
*ch ioldr(f) is
given is misguiding in that it does not consist of monomials of that form. To g*
*et it into
the desired form, we would take ioldr(f(t0)) and break it up, using Fourier exp*
*ansion, into
monomials of the form
g0(t0) a1tn10 a2tn20 . . .artnr0= g0(t0)tn1+n2+...nr0 a1 a2 . . .ar
(using the fact that t0 is just a complex number which can pass through the ten*
*sor). This
presentation makes it clear why, on the image of ioldr, dfr_dt= 0.
It remains to show that inewrOssnewr' id, by finding a suitable homotopy Kn*
*ew:.
As motivation for the construction, note that we can view A[t; t-1] as embedded*
* into
C1 [S1; A], by looking at t as the inclusion map S1 ,! C.1A and at t-1 as the c*
*omposition
t7!1=t
S1----! S1 ,! C . 1A . By this map, the (regular) tensor products of A[t; t-1] *
*with itself
embed into the topological projective tensor products of C1 [S1; A] with itself*
*, and all the
constructions and maps we have defined commute with these embeddings. Thus to *
*find
the homotopy Knew:for C1 [S1; A] we adapt the homotopy which was used for A[t; *
*t-1].
P For any C1 function f on the circle, we can use the Fourier expansion f(t*
*) =
1 n
n=-1 ant to write
f = f- + a0 + f+
where -1
X X1
f- (t) = antn ; f+ (t) = antn :
n=-1 n=1
This is necessary because of the asymmetry between t and t-1 in the formulae of*
* the
previous section. We set
(K newr(f0 f1 . . .fr))(t0; t1; : :;:tr; tr+1)
Xr j-1X h
= (-1)h+1 f0(t0) f1(t0) . . .fh(t0) th+1 fh+1(t0) . . .fj-1(t0)
j=1h=0
f+j(t0)- f+j(tj+1)
_________________t fj+1(tj+2) . . .fr(tr+1)
0 - tj+1
- f0(t0) f1(t0) . . .fh(t0) th+1 fh+1(t0) . . .fj-1(t0)
f-j(t0)- f-j(tj+1) i
_________________t fj+1(tj+2) . . .fr(tr+1)
0 - tj+1
r-1X X`
+ dfr_dt(t0) (-1)r-h b(a)0 . . .b(a)h th+1 b(a)h+1 . . .b(a)r
h=0 a=0
+ Koldr(f0(t0) f1(t0) . . .fr(t0))
13
where
X`
Koldr-1(f0(t0) f1(t0) . . .fr-1(t0)) = b(a)0 . . .b(a)1 . . .b(a)r :
a=0
This definition coincides with (2.3) on monomials f0 f1 . . .fr where_fi(t)_*
*= aitni
for all 0 i r and t 2 S1, so we_know_that_on the copy of A[t; t-1] A[t; t-1]*
*(r)which
is embedded inside C1 [S1; A] C1 [S1; A]^ss(r)we have
(3:4) dr+1OKnewr+ Knewr-1Odr = id- inewrOssnewr :
Note however that all the functions which appear in equation (3.4) are continu*
*ous as
functions from the reduced bar complex to itself (with respect to the norm on e*
*ach space
C1 (Tr+1; A ^ssA ^ss. .^.ssA)= [r-1i=0Im(si)which is induced by the supremum no*
*rm on
C1 (Tr+1; A_^ssA_^ss._.^.ssA)). Fourier expansion shows us that the_embedded_c*
*opy of
A[t; t-1] A[t; t-1](r)on which (3.4) holds is dense_in_C1_[S1; A] C1 [S1; A]^*
*ss(r). We
deduce that (3.4) holds on all of C1 [S1; A] C1 [S1; A]^ss(r), and so Knew:is*
* indeed the
desired homotopy.
REFERENCES
[1]C. Kassel, Homologie cyclique, caractere de Chern, et lemme de perturbation*
*, J. Reine
u. Ang. Math. 408 (1990) 159-180.
[2]J.-L. Loday, Cyclic homology, Grundlehren der mathematischen Wissenshaften *
*301,
Springer Verlag 1992.
[3]K. Wolffhardt, The Hochschild homology of complete intersections, Trans. A*
*mer.
Math. Soc. 171 (1972) 51-66.
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