## Excision in algebraic $$K$$-theory.(English)Zbl 0756.18008

Suppose $$R$$ is a ring with unit. For $$i\geq 1$$, the algebraic $$K$$-theory groups $$K_ i(R)$$ are defined to be $$\pi_ i(\text{BGL}(R)^ +)$$, the homotopy groups of the Quillen plus construction applied to the space $$\text{BGL}(R)$$. Rational algebraic $$K$$-theory is defined analogously using homotopy with rational coefficients. If $$A\subset R$$ is a 2-sided ideal, the relation $$K$$-groups $$K_ i(R,A)$$, $$i\geq 1$$, are the homotopy groups of the homotopy fiber, $$F(R,A)$$ of the map $\text{BGL}(A)^ +\to B(\text{Im}(\text{GL}(R)\to\text{GL}(R/A)))^ +,$ giving rise to a functorial long exact sequence in algebraic $$K$$-theory.
Now suppose $$A$$ is a ring without unit. We define $$K_ i(A)=K_ i(\widetilde A,A)$$, where $$\widetilde A$$ is the unital ring obtained by adjoining a unit to $$A$$. (This agrees with the previous definition when $$A$$ already has a unit.)
A ring $$A$$ satisfies excision in algebraic $$K$$-theory if, whenever $$R$$ is a unital ring containing $$A$$ as a 2-sided ideal, $$K_ *(A)\to K_ *(R,A)$$ is an isomorphism (i.e. the $$K$$-theory of $$A$$ is independent of the particular embedding of $$A$$ into a unital ring). Determining which rings satisfy excision is an old problem in algebraic $$K$$-theory. The authors’ main theorem asserts the equivalence of (a) $$A$$ satisfies excision in rational algebraic $$K$$-theory; and (b) $$A\otimes_{\mathbb{Z}}\mathbb{Q}$$ is $$H$$-unital. Here a $$\mathbb{Q}$$-algebra $$B$$ is said to be $$H$$-unital if the chain complex $A@< b'<< B\otimes_{\mathbb{Q}} B@< b'<< B\otimes_{\mathbb{Q}} B\otimes_{\mathbb{Q}} B@< b'<<\cdots$ is acyclic, where $b'(b_ 1\otimes\cdots\otimes b_ q)=\sum_ i(- 1)^{i-1} b_ 1\otimes\cdots\otimes b_ i b_{i+1}\otimes\cdots\otimes b_ q.$ For $$\mathbb{Q}$$-algebras $$A$$, the main theorem implies that the $$H$$-unitality of $$A$$ is equivalent to excision in algebraic $$K$$-theory. All $$C^*$$-algebras are proved to be $$H$$-unital; this implies the conjecture of Karoubi that topological and algebraic $$K$$-theory coincide on the category of stable $$C^*$$- algebras.
In all cases for which the authors prove excision in algebraic $$K$$- theory, they actually prove the stronger assertion: $$\text{BGL}(A)^ +$$ is an infinite loop space and the canonical map $$\text{BGL}(A)^ +\to F(R,A)$$ is a homotopy equivalence for any unital ring $$R$$ containing $$A$$ as a 2-sided ideal.

### MSC:

 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19D55 $$K$$-theory and homology; cyclic homology and cohomology 55P35 Loop spaces 19D06 $$Q$$- and plus-constructions 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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