Cohomologie cyclique et foncteurs \(Ext^ n\).

*(French)*Zbl 0534.18009In his articles ”Non-commutative differential geometry. I, II” (I.H.E.S. preprints), the author has defined and studied the cyclic cohomology \(H^ n_{\lambda}({\mathcal A})\) of a \(C^*\)-algebra. The purpose of this note is to show how that cohomology may be obtained in terms of the Ext functor, where the ”modules” in question are functors from a suitable small category.

To be specific, the author defines a category \(\Lambda\) whose objects \(\Lambda_ n\) are indexed by the non-negative integers and whose morphisms \(f\in Hom(\Lambda_ n,\Lambda_ m)\) are homotopy classes of continuous increasing maps \(\phi\) : \(S^ 1\to S^ 1\) which map the \((n+1)th\) roots of unity into the \((m+1)th\) roots of unity. A \(\Lambda\)- module is then a covariant functor from \(\Lambda\) into the category of Abelian groups, and, for a field k, a \(k(\Lambda)\)-module is a covariant functor from \(\Lambda\) into the category of vector spaces over k. He associates to each unital algebra \({\mathcal A}\) over k a \(k(\Lambda)\)-module \({\mathcal A}^{\diamond}\) whose objects are the n-fold tensor products of \({\mathcal A}\) with itself. A similar definition is made when \({\mathcal A}\) is a ring; in this case \({\mathcal A}^{\diamond}\) is a \(\Lambda\)-module. The author shows that for a k-algebra \(H^ n_{\lambda}({\mathcal A})\) is just \(Ext^ n_{k(\Lambda)}({\mathcal A}^{\diamond},k^{\diamond}).\)

To be specific, the author defines a category \(\Lambda\) whose objects \(\Lambda_ n\) are indexed by the non-negative integers and whose morphisms \(f\in Hom(\Lambda_ n,\Lambda_ m)\) are homotopy classes of continuous increasing maps \(\phi\) : \(S^ 1\to S^ 1\) which map the \((n+1)th\) roots of unity into the \((m+1)th\) roots of unity. A \(\Lambda\)- module is then a covariant functor from \(\Lambda\) into the category of Abelian groups, and, for a field k, a \(k(\Lambda)\)-module is a covariant functor from \(\Lambda\) into the category of vector spaces over k. He associates to each unital algebra \({\mathcal A}\) over k a \(k(\Lambda)\)-module \({\mathcal A}^{\diamond}\) whose objects are the n-fold tensor products of \({\mathcal A}\) with itself. A similar definition is made when \({\mathcal A}\) is a ring; in this case \({\mathcal A}^{\diamond}\) is a \(\Lambda\)-module. The author shows that for a k-algebra \(H^ n_{\lambda}({\mathcal A})\) is just \(Ext^ n_{k(\Lambda)}({\mathcal A}^{\diamond},k^{\diamond}).\)

Reviewer: W.Moran

##### MSC:

18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |

46L05 | General theory of \(C^*\)-algebras |

46M05 | Tensor products in functional analysis |

16Exx | Homological methods in associative algebras |