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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 2

### The algebraic $K$-theory of a diagram of rings

Pages: 13 – 58

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n2.a2

#### Author

#### Abstract

In this paper, we consider “diagrams of rings”, or functors from a small category to the category of rings, and the corresponding diagrams of groups $K_i.$ Classically, this was initiated by Milnor. The main result of this paper is the direct comparison of the filtration in classical algebraic $K$-theory discussed in J. Duflot, “Simplicial groups that are models for algebraic $K$-theory,” *Manuscripta Math.* 113 (2004), no. 4, 423-470 and J. Duflot and C.T. Marak, “A filtration in algebraic $K$-theory,” *J. Pure Applied Algebra* 151 (2000), no. 2, 135-162 to a corresponding filtration in the Bousfield-Kan spectral sequence associated to a *Tot*-tower of simplicial groups attached to the diagram of rings.

#### Keywords

algebraic $K$-theory; simplicial group

#### 2010 Mathematics Subject Classification

18G30, 18G55, 19Dxx, 55U10