## Algebraic $$K$$-theory of parameterized endomorphisms.(English)Zbl 1115.18005

For a bimodule $$M$$ over a ring $$R$$ let End$$(R,M)$$ denote the following category, called the category of parameterized endomorphisms: objects are pairs $$(P,f)$$ with $$P$$ a finitely generated projective right $$R$$-module and $$f:P \rightarrow P \otimes M$$ a homomorphism, and morphisms $$\Phi: (P,f) \rightarrow (Q,g)$$ are given by maps $$\phi: P \rightarrow Q$$ satisfying $$g \circ \phi = (\phi \otimes id) \circ f.$$ Assume now that $$R$$ is a commutative semi-simple ring. Let $$TM = \oplus_{i \geq 0}M^{\otimes i}$$ denote the tensor algebra on $$M$$. Any object $$(P,f) \in \text{End}(R,M)$$ induces a map $$id - f: P\otimes TM \rightarrow P\otimes TM$$, and $$(P,f)$$ is nilpotent if and only $$id-f$$ is an isomorphism. Let $$Nil$$ denote the subcategory of End$$(R,M)$$ of all nilpotent objects. The author deduces from results of M. Schlichting [cf. “Delooping the $$K$$-theory of exact categories”, Topology 43, 1089–1103 (2004; Zbl 1059.18007)]) that there is a homotopy fibration of $$K$$-theory spaces $K(Nil) \rightarrow K(\text{End}(R,M)) \rightarrow K(\mathcal H),$ where $$\mathcal H$$ is obtained from End$$(R,M)$$ by formally inverting weak isomorphisms. It is shown that $$\mathcal H$$ is equivalent to the full subcategory $$H(TM,E)$$ of the category of right $$TM$$-modules, whose objects are cokernels of the maps $$id-f: P \otimes TM \rightarrow P\otimes TM$$ with $$(P,f) \in \text{End}(R,M)$$. As a consequence one obtains a long exact sequence of algebraic $$K$$-theory groups $\cdots \rightarrow K_{i+1}(H(TM,E)) \rightarrow K_i(Nil) \rightarrow K_i(\text{End}(R,M)) \rightarrow K_i(H(TM,E)) \rightarrow \cdots$ In the special case that $$R$$ is a field the author relates the $$K$$-theory of $$H(TM,E))$$ to the $$K$$-theory of the noncommutative localization $$\sigma^{-1}TM$$ of $$TM$$ with respect to the collection $$\sigma$$ of $$TM$$-maps $$id-f: P \otimes TM \rightarrow P\otimes TM$$, and uses the localization sequence of A. Neeman and A. Ranicki, “Noncommutative localization in algebraic $$K$$- theory. I” [Geom. Topol. 8, 1385-1425 (2004; Zbl 1083.18007)] to show that one obtains isomorphisms $\tilde{K}_n(\text{End}(R,M)) = \tilde{K}_{n+1}(\sigma^{-1}TM)$ between the reduced $$K$$-groups.

### MSC:

 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)

### Citations:

Zbl 1059.18007; Zbl 1083.18007
Full Text:

### References:

 [5] Neeman, A. and Ranicki, A.: Noncommutative localization and chain complexes I. Algebraic K- and L-theory. Preprint. · Zbl 1083.18007 [6] Quillen, D.: Higher Algebraic K-theory: I, Lecture Notes in Math. 341 (1972), · Zbl 0249.18022 [8] Schofield, A. H.: Representations of Rings Over Skew Fields. LMS Lecture Note Series 92, Cambridge University Press, Cambridge, 1985.
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