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Bivariant cyclic theory. (English) Zbl 0755.18008

Authors’ introduction: “The purpose of this article is to study a bivariant form of cyclic cohomology \(HC^*(A,B)\) defined for any pair of unital associative \(k\)-algebras where \(k\) is a commutative ring. This bivariant theory is a \(k\)-module, graded by the integers \(\mathbb{Z}\) and it has the following fundamental properties:
Bivariance. \(HC^*(A,B)\) is a contravariant functor of \(A\) and a covariant functor of \(B\).
Specialisation. There are natural isomorphisms \(HC^ p(A,k)\cong HC^ p(A)\), \(HC^ p(k,B)\cong HC^ -_{-p}(B)\), where \(HC^*(A)\) is the cyclic cohomology of \(A\), defined by A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 257-360 (1985; Zbl 0592.46056)], and \(HC^ -_ *(B)\) is the negative cyclic homology of \(B\) defined by T. G. Goodwillie [Ann. Math., II. Ser. 124, 347-402 (1986; Zbl 0627.18004)] and the first author [Invent. Math. 87, 403-423 (1987; Zbl 0644.55005)].
Product. Let \(A_ 1\), \(A_ 2\), \(B_ 1\), \(B_ 2\), \(D\) be a unital \(k\)- algebras. Then there is a natural associative pairing \[ HC^*(A_ 1,B_ 1\otimes D)\otimes HC^*(D\otimes A_ 2,B_ 2)\to HC^*(A_ 1\otimes A_ 2,B_ 1\otimes B_ 2) \] which extends the products on cyclic cohomology and negative cyclic homology.
The main idea in our approach to bivariant cyclic theory is to exploit the fact that cyclic homology and cohomology groups are completely determined by the Hochschild complex, \(C_ *(A)\), the Hochschild boundary operator \(b: C_ n(A)\to C_{n-1}(A)\) and Connes’ coboundary operator \(B: C_ n(A)\to C_{n+1}(A)\). These are defined as follows: \(C_ n(A)=A^{\otimes(n+1)}\), \[ b(a_ 0\otimes\cdots\otimes a_ n)= \]
\[ \sum^{n-1}_{i=0}(-1)^ ia_ 0\otimes\cdots\otimes a_ i a_{i+1}\otimes\cdots\otimes a_ n+(-1)^ n a_ n a_ 0\otimes a_ 1\otimes\cdots\otimes a_{n-1}, \]
\[ B(a_ 0\otimes\cdots\otimes a_ n)= \]
\[ \sum^ n_{i=0}(-1)^{ni}(1\otimes a_ i\otimes\cdots\otimes a_ n\otimes a_ 0\otimes\cdots\otimes a_{i-1}+ \]
\[ (-1)^ n a_ i\otimes\cdots\otimes a_ n\otimes a_ 0\otimes\cdots\otimes a_{i- 1}\otimes 1). \] The operators \(b\) and \(B\) satisfy the identities \(b^ 2=B^ 2=bB+Bb=0\).
We will study this general structure in the following way. Let \(\Lambda\) be the exterior algebra over \(k\) on one generator \(\varepsilon\) of degree 1. The \(B\)-operator makes \(C_ *(A)\) into a graded module over \(\Lambda\) by setting \(\varepsilon x=Bx\). The \(b\) operator provides \(C_ *(A)\) with a differential and in this way \(C_ *(A)\) becomes a differential graded \(\Lambda\)-module (dg-\(\Lambda\)-module). Moreover, regarding \(C_ *(A)\) as a dg-\(\Lambda\)-module retains precisely the minimum structure necessary to define cyclic homology and cohomology.
We can do differential homological algebra in the category of dg- \(\Lambda\)-modules. So let \(M\), \(N\) be dg-\(\Lambda\)-modules, then we define \(HC^*(M,N)\) as follows: \(HC^*(M,N)=\text{Ext}^*_ \Lambda(M,N)\). If \(A\) and \(B\) are unital \(k\)-algebras then we define the bivariant cyclic groups to be \(HC^*(A,B)=HC^*(C_ *(A), C_ *(B))\); these are the bivariant cyclic groups mentioned above.
The natural way to compute Ext-groups is to use the bar construction. In fact the bar construction on the dg-\(\Lambda\)-module \(C_ *(A)\) corresponds precisely to Connes’ \((b,B)\) bicomplex for computing \(HC_ *(A)\). This relation between the bar complex and the \((b,B)\)-bicomplex is one of the key points in this approach to bivariant cyclic theory.
In Section 1, we develop some necessary preliminaries involving the bar construction and in Section 2 we explain how the bar construction, in other words the \((b,B)\)-bicomplex, gives an explicit chain complex whose homology is the bivariant theory \(HC^*(M,N)\). In Section 3, we construct some natural elements in the bivariant groups \(HC^*(A,B)\) and, in Section 4, we prove universal coefficient theorems expressing the bivariant groups in terms of cyclic homology and negative cyclic homology. In Section 5, we describe the product in the bivariant cyclic theory and, in Section 6, we exploit the product structure in the bivariant groups to analyse the notion of an \(HC\)-equivalence.
In Section 7 we show how to compute bivariant cyclic groups involving the algebras \({\mathcal O}(X)\) and \({\mathcal D}(X)\) of regular functions and differential operators on a smooth affine algebraic variety over a field of characteristic zero. This amounts to assembling various calculations in the literature.
There is a natural periodicity operator \(S: HC^ p(M,N)\to HC^{p+2}(M,N)\), and we define periodic bivariant theory \(HP^*(M,N)\) to be the direct limit of the system \[ \cdots@> S >> HC^ p(M,N)@> S >> HC^{p+2}(M,N)@> S >>\cdots. \] In Section 8 we show how to compute this periodic bivariant cyclic theory in terms of the periodic cyclic theory of \(M\) and \(N\).
Another approach to bivariant cyclic theory is to use Connes’ theory of cyclic \(k\)-modules. The main difference between the two approaches is that bivariant groups in the category of cyclic \(k\)-modules are defined using classical homological algebra rather than the differential homological algebra we use here. In Sections 9 and 10 we begin the project of comparing these two approaches to bivariant theory. While our results in this direction are far from complete we do at least show that the \(HC^-\)-groups can be expressed as Ext-groups, in the sense of classical homological algebra, in the category of cyclic \(k\)-modules”.

MSC:

18G60 Other (co)homology theories (MSC2010)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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References:

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