Excision in entire cyclic cohomology
The author generalizes Connes' definition of entire cyclic cohomology \(HE^*\) to bornological algebras. By a bornological algebra is meant a (convex) bornological vector space with a bounded associative multiplication. If \(A\) is an algebra, then Cuntz and Quillen's approach to periodic cyclic homology yields a complex which is denoted \(X(\mathcal TA)\). If \(A\) is a bornological algebra the author gives \(X(\mathcal TA)\) a bornology, and defines the entire cyclic cohomology \(HE^*(A)\) as the complex of bounded linear maps \(X(\mathcal TA)\to\mathbb{C}\). NEWLINENEWLINEAs the title indicates, the paper is devoted to showing the following excision result. Let NEWLINE\[NEWLINE0\longrightarrow K\overset{i}\longrightarrow E\overset{p}\longrightarrow Q\longrightarrow 0NEWLINE\] NEWLINEbe an extension of bornological algebras with a bounded linear section \(Q\to E\). Then the bornological complex \(X(\mathcal TE:\mathcal TQ)\) calculating the relative entire cyclic cohomology is naturally homotopy equivalent to \(X(\mathcal TK)\). It follows that there is a natural six-term exact sequence NEWLINE\[NEWLINE\begin{tikzcd} NEWLINEHE^0(Q)\ar[r,"p^\ast"] & HE^0(E)\ar[r,"i^\ast"] & HE^0(K)\ar[d]\\ NEWLINEHE^1(K)\ar[u] & HE^1(E)\ar[l,"i^\ast" '] & HE^1(Q)\ar[l,"p^\ast" ']\rlap{\,.} NEWLINE\end{tikzcd}NEWLINE\]NEWLINEThe proof also yields similar results for the associated bivariant and periodic theories. NEWLINENEWLINEThe proof presented in this paper differs from the proofs of similar statements as by \textit{J. Cuntz} [in: Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, 43-53 (1997; Zbl 0890.18006)], \textit{J. Cuntz} and \textit{D. Quillen} [Invent. Math. 127, No. 1, 67-98 (1997; Zbl 0889.46054)], \textit{M. Puschnigg} [Invent. Math. 143, No. 2, 249-323 (2001; Zbl 0973.19002)] in that it does not rely on the methods of \textit{M. Wodzicki} [Ann. Math., II. Ser. 129, No. 3, 591-639 (1989; Zbl 0689.16013)] which are ``difficult to adapt to entire cyclic cohomology''. NEWLINENEWLINEInstead the author shows that \(X(\mathcal TK)\to X(\mathcal TE:\mathcal TQ)\) is a homotopy equivalence of (filtered) bornological complexes by introducing the left ideal \(\mathcal L\) in \(\mathcal TE\) generated by \(K\), and then showing that both \(X(\mathcal L)\to X(\mathcal TE:\mathcal TQ)\) and \(X(\mathcal TK)\to X(\mathcal L)\) are homotopy equivalences. The first homotopy equivalence is established by using homological algebra to show that \(X(\mathcal TE:\mathcal TQ)\) is isomorphic to a sum \(X(\mathcal L)\oplus C\) where \(C\) is contractible. The second is obtained by constructing an explicit homotopy inverse. Excision then follows for bivariant entire and periodic cyclic cohomology.
scientific article; zbMATH DE number 1656347