Higher Algebraic K-Theory

Introduction Orthogonal spectra and $S$-modules Equivariant orthogonal spectra Model categories of orthogonal $G$-spectra Orthogonal $G$-spectra and $S_G$-modules ""Change"" functors for orthogonal $G$-spectra ""Change"" functors for $S_G$-modules and comparisons Bibliography Index of notation.

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Equivariant Orthogonal Spectra and S-Modules

Intersection theory on algebraic stacks and Q-varieties

We construct a functor KG which takes each pair of monoidal G-graded categories (D,Df) to an infinite loop G-space KG(D,D'). When D'=D, its homotopy groups n%KG(D,D) coincide with the equivariant K-groups KnRepD of D. Applications include the simple construction of equivariant infinite deloopings of the maps BO(G}—^BPL(G}-^> BTop(G} between equivariant classifying spaces.

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=25&iss=2&rank=3

Infinite Loop G-Spaces Associated to Monoidal G-Graded Categories

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V–fold loop G–spaces to several avatars of a recognition principle for infinite loop G–spaces. We then explain what genuine permutative G–categories are and, more generally, what E∞–G–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine G–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension G–spectra. Other examples are geared towards equivariant algebraic K–theory.

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Equivariant iterated loop space theory and permutative G–categories

There are many ways that group actions enter into algebraic K-theory and there are various theories that fit under the rubric of our title. To anyone familiar with both equivariant topological K-theory and Quillen's original definition and calculations of algebraic K-theory, there is a perfectly obvious program for the definition of the equivariant algebraic K-theory of rings and its calculation for finite fields. While this program surely must have occurred to others, there are no published accounts and the technical details have not been worked out before. That part of the program which pertains to the complex Adams conjecture was outlined in a letter to one of us from Graeme Segal, and the real analog was assumed without proof by tom Dieck [10,11.3.8]. Negatively indexed equivariant K-groups were introduced by Loday [19]. From a topological point of view, one way of thinking about Quillen's original definition runs as follows. Let ~ be a topological group, perhaps discrete. One has a notion of a principal K-bundle and a classifying space B~ for such bundles. When H is discrete, a principal ~-bundle is just a covering (possibly with disconnected total space) with fibre and group ~. Given any increasing sequence of groups N with union H, we obtain an increasing sequence of classifying spaces n BH with union B~. We then think of B~ as a classifying space for stable n bundles. When the ~ are discrete, B~ may have desirable homology groups but n will have trivial higher homotopy groups. In the cases of interest, we can use the plus construction to convert B~ to a Hopf space (= H-space) (BN) + with the same homology. When ~n = GL(n,A), we define ~q(B~) + = Kq(A) for q > 0. There is a more structured way of looking at (B~) +. In practice, we have sum maps {9 :~ x ~ ÷ ~m+n and a corresponding Whitney sum of bundles. While this can m n he used to give B~ a product, it is generally not a Hopf space, although it is so

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H. Hauschild

Papers

Equivariant algebraic K-theory