Deformation Retracts and the Hochschild Homology
of Polynomial Rings
AyeletLindenstrauss*
Universityof Pennsylvania
Philadelphia, PA 19104
x0. Introduction
In computations of Hochschild homologyand its variants, such as cyclic homo*
*logy,
neither the standard Hochschildcomplex
d3!A A A d2!A Ad1! 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, wh*
*ich 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=k0! n1A=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 ^ ^ dfr7! 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 fk7! f0df1 ^ ^ dfr :
!
!
Clearly!ssffii!= idrA=k, and by the Hochschild-Kostant-Rosenberg theorem,i and *
*ss are
actually!quasi-isomorphisms. Following Kassel [1], we define adeformation retra*
*ct (of the
reduced!Hochschild!complex) to be a complex (X:;dX ) together with chain maps i*
*:: X:!
A!!A() andss: : A A() ! X: and achain homotopy K:: A A() ! A A(+1)
such!that
!
!
!! ssrffiir = idXr ; dr+1ffiKr+ Kr1 ffidr =idAA(r) irffissr :
!
* Partially supported by N.S.F. Grant No. DMS 92-03398
Hitherto,there have been ad hoc computations of deformation retracts, but n*
*oattempt
to systematize the construction. This pap eris 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;t1 ]. The latter construction can be adapted to give,for each Banach al*
*gebra A
and deformation retract of the continuous reduced Hochschild complex for A,a de*
*formation
retract for the ring C1 [S1;A] of smooth A-valued functions on the circle. It s*
*eems likely
that similar methods will work for the non-commutative polynomial rings
Afxg=(ax xOE(a)) ;
where OE is an automorphism of A. Amore difficult question, which we donot disc*
*uss, is
the construction of deformation retractsfor A=a A, where a is an element of th*
*e 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 th*
*e state of
the art in computing Hochschild homology since the Kunneth formula already exp*
*resses
HH (A[x]) and HH (A[t;t1 ]) in terms of HH (A). The results which follow do,*
* however,
provide information about explicit homotopies that the Kunneth formula does no*
*t. Already
in the case of k[x;y], a homotopy contracting the reduced Hochschild bar comple*
*x to the
Hochschild-Kostant-Rosenberg complex is quite difficult to find. The methods of*
* this paper
give a general inductive construction.