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Excision in cyclic homology and in rational algebraic K-theory. (English) Zbl 0689.16013
Let $$F_* = \{F_ q,\;q\geq 0\}$$ be a sequence of functors from the category of rings without unit to, say, the category of abelian groups. A ring $$A$$ (without unit) is said to have excision for $$F_*$$ if, for any ring extension $$A\rightarrowtail R\twoheadrightarrow S$$, there is a natural long exact sequence $$\to F_{q+1}(S) \to F_ q(A) \to F_ q(R) \to F_ q(S) \to \ldots$$. This means that $$F_ q(A)$$ depends only on $$A$$, not on the ambient ring $$R$$, and it implies that for a Cartesian square $\begin{tikzcd} P \rar\dar & R\ar[d,"f"] \\Q \rar & S\rlap{\,,} \end{tikzcd}$ there is a Mayer-Vietoris long exact sequence $\ldots \to F_{q+1}(S) \to F_ q(P) \to F_ q(Q) \oplus F_ q(R) \to F_ q(S) \to \ldots$ It is well-known that excision fails for the algebraic $$K$$-theory functors $$K_*$$.
The author shows that, if a ring $$A$$ has excision for rational algebraic $$K$$-theory $$\mathbb{Q}\otimes K_*$$, then the $$\mathbb{Q}$$-algebra $$A\otimes\mathbb{Q}$$ has excision for cyclic homology. He conjecures that the reverse implication also holds. (There are underlying technical hypotheses that all rings are $$k$$-algebras for some commutative algebra $$k$$ and that all ring extensions are pure as $$k$$-module extensions.)
The author gives a complete characterization of rings that have excision for cyclic homology as follows. Let $$B_*(A)$$ be the chain complex $\{B_ q(A) = A^{\otimes q}\}, \quad b'(a_1\otimes\cdots\otimes a_ q\} = \sum(-1)^{i-1}a_1\otimes\cdots\otimes a_ i a_{i+1}\otimes \cdots\otimes a_ q.$ If $$A$$ has a unit, this is the standard Bar resolution and the complex $$B_*(A)\otimes V$$ has trivial homology for any $$k$$-module $$V$$. In general, $$A$$ is said to be $$H$$-unital if $$B_*(A)\otimes V$$ has trivial homology. Then $$A$$ is $$H$$-unital if and only if $$A$$ has excision in cyclic homology. This equivalence also holds for Hochschild homology.
The author then develops criteria for determining H-unital algebras and gives many examples to show that the concept has wide applicability. Among these examples are: a right ideal in a simple ring; the $$\mathbb{Z}$$- algebra of $$C^\infty$$-functions with compact support on a $$C^\infty$$- manifold; locally convex algebras, ideals of flat $$C^\infty$$-functions and of flat differential operators on a smooth manifold; Banach algebras with bounded approximate units. Finally, the author shows that ‘$$H$$- unitality’ is preserved by various constructions. The notion of an $$H$$- unital module is introduced, and used to show that a matrix algebra over an $$H$$-unital ring is again $$H$$-unital. If $$A$$ is $$H$$-unital, so are the cone and suspension of $$A$$. If $$B$$ is also $$H$$-unital and $$M$$ is an $$A$$- $$B$$-bimodule, the triangular ring $$\begin{pmatrix} A & M \\ 0 & B \end{pmatrix}$$ is $$H$$-unital.
Reviewer: M.E.Keating

##### MSC:
 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16E20 Grothendieck groups, $$K$$-theory, etc. 18G35 Chain complexes (category-theoretic aspects), dg categories 46L80 $$K$$-theory and operator algebras (including cyclic theory) 16S20 Centralizing and normalizing extensions
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