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.


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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