Excision in cyclic homology and in rational algebraic K-theory.

*(English)*Zbl 0689.16013Let \(F_* = \{F_ q,\;q\geq 0\}\) 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.

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 |