Holmstrom 5 Memo notes Carlsson and Hesselholt

Memo notes from Carlsson: Deloopings in Algebraic K-theory

The full article is found here

Quillen’s insight: Should define K-groups as homotopy groups of a space. Two methods: plus construction and Q construction. This functor, apparently from rings/schemes to Top, actually is a functor to the category of infinite loop spaces and infinite loop maps, or (perhaps even better) to the category of spectra. Various advantages of this viewpoint: homotopy colimits, lower K-groups of Bass, spectrum homology, a “linearity” property: fibre of a map is equivalent to the loop spectrum of the cofibre.

Will give an overview of ways of attaching spectra to rings, i.e. deloopoing the K-theory space of a ring. Outline: A generic construction. The Q-construction. Iterations of the Q-con, by Waldhausen, Gillet-Grayson, Jardine, Shimakawa. Waldhausen’s S-construction. Non-connective delopings by Gersten-Wagoner, Karoubi, Pedersen-Weibel.

Generic deloopings using infinite loop space machines

Eilenberg-MacLane spaces/spectra attached to abelian groups.

Segal’s Gamma spaces: Defined as a contravariant functor from a certain category $\Gamma$ to $Sset$, satisfying some conditions. Every $\Gamma$-space can be viewed as a bisimplicial set.

For any category with a zero objects and finite direct sums, we construct a $\Gamma$-space. This is done through a functor from $\Gamma^{op}$ to $Cat$, which we compose with the nerve construction. This construction “behaves up to homotopy like the classifying space construction for abelian groups”. Generalise this to produce a functor $Sp$ from cats with zero and sum to the category of spectra. Examples: The category of finite sets gives the sphere spectrum, and the category of f.g. projective modules over a ring $A$ gives the K-theory spectrum of $A$.

Would like to work with more general cats $C$ (i.e. without zero and direct sums), to include for example the Eilenberg-MacLane spectrum of an abelian group. This was done by May and Thomason, to produce a functor from symmetric monoidal cats to spectra. For permutative cats, i.e. symmetric monoidal ones where the associativity isomorphism is the identity, May used operads instead of $\Gamma$-spaces to obtain the same result.

The Q-construction and its higher-dimensional generalizations

Quillen defined, for any exact category $C$, another exact category $Q(C)$, and $K_{i-1}(C)$ as the group $\pi_i N(Q(C))$. Want: higher deloopings, i.e. spaces $X_n$ such that $K_{i-n}(C) = \pi_i X_n$. (This is a bit weird, as deloopings become higher we have to use a higher homotopy group to define a fixed K-group. Are the indices wrong??)

To obtain this, Shimakawa uses multicategories (1986), generalizing the Q construction and the nerve.

Waldhausen’s S-construction

Input: category with cofibrations and weak equivalences (much more general than exact categories, for example, can consider various cats of spaces).

Def: Category with cofibs and WEs. Exact functor between such cats.

Examples: The category of based finite sets. The category of pointed simplicial sets. Any exact category, where the cofibs are taken to be the admissible monomorphisms, and the WEs are the IMs. Various cats of chain complexes.

Have localization and additivity theorems generalizing those of Quillen.

Notions of cylinder functor, saturation axiom, extension axiom.

The Gersten-Wagoner delooping

Nonconnective delooping, on the level of rings…

Deloopings based on Karoubi’s derived functors

Flasque additive category? Idempotent complete additive category (Schlichting)?

The Pedersen-Weibel Delooping and Bounded K-theory

Construction of the “bounded K-theory spectrum of a metric space $X$ with coefficients in a ring $A$” . Applications to the Novikov conjecture.

Memo notes from Hesselholt in K-theory handbook

K-theory of truncated polynomial algebras.

Intro

Given a ring $A$ and a two-sided ideal $I$, can define relative K-theory, so that there is a natural exact triangle of spectra

and hence a corresponding l.e.s. (going downwards).

If the ideal is nilpotent, these relative groups can be expressed in terms of cyclic homology and topological cyclic homology (refs on this). Arguments use Goodwillie’s calculus of functors.

Some ingredients: Absolute differential forms, big de Rham-Witt differential forms, the Witt complex, pro-objects.

Topological Hochschild homology

Def of $THH_*(A)$. Alternative definition: The homology groups of the category of finitely generated projective $A$-modules, with coefficients in the bifunctor $Hom$. (Jibladze-Pirashvili)

Rest of article omitted. Some references which did not fit in a CT page:

Hesselholt and Madsen: On the de Rham-Witt complex in mixed characteristic (2004).

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