Tags: #higher-algebra/K-theory #projects/notes/reading #motivic/schemes

# Algebraic K-Theory for schemes

## References

Reference:Sheaves for Homotopy Theorists

## Notes

The algebraic \({\mathsf{K}}{\hbox{-}}\)groups of \(X\in {\mathsf{Sch}}\) are defined by producing a __spectrum__ \({\mathsf{K}}(X) \in {\mathsf{Sp}}\) and setting \(K^n(X) \coloneqq\pi_{-n}K(X)\).Be warned that \({\mathsf{K}}{\hbox{-}}\)theorists and geometers write \({\mathsf{K}}_n(X)\) for our \({\mathsf{K}}^{−n}(X)\) – we will stick to homotopy-theoretic notation, however.

\({\mathsf{K}}\) can be defined so that it actually gives a functor \({\mathsf{Sch}}^^{\operatorname{op}}\to {\mathsf{Sp}}\) and so the groups \({\mathsf{K}}^n(X)\) start to look like a cohomology theory. For \(X\in {\mathsf{Top}}\), the analogue of \({\mathsf{K}}(X)\) turns out to be the __mapping spectrum__ \(\bu^X\), where __bu__ is the spectrum representing complex connective K-theory (so that \(\bu\) is the __connective cover__ of the spectrum \({\mathbf{B}}{\operatorname{U}}\)).

Thus, the functor \({\mathsf{K}}\) can be thought of as a substitute for the spectrum \(\bu\) – it is sort of a device for storing all the same information that bu provides, but without an honest ‘space’ to house it in.

Most of the important results about __Algebraic K theory__ arise from the study of the spectrum \({\mathsf{K}}(X)\), rather than that of the disembodied abelian groups \({\mathsf{K}}^n(X)\). For example, if the scheme \(X\) is covered by two open sets \(U\) and \(V\) , one wants a Mayer-Vietoris sequence
\begin{align*}
\cdots \rightarrow K ^ { n } ( X ) \rightarrow K ^ { n } ( U ) \oplus K ^ { n } ( V ) \rightarrow K ^ { n } ( U \cap V ) \rightarrow K ^ { n + 1 } ( X ) \rightarrow \cdots
\end{align*}

This follows formally once one proves the stronger result that \begin{align*} \begin{CD} \mathcal{K}(C) @>>> \mathcal{K}(U) \\ @VVV @VVV \\ \mathcal{K}(V) @>>> \mathcal{K}(U\cap V) \end{CD} \end{align*} is a homotopy pullback diagram.

Our first guess is motivated by the hypothesis that a functor \(E: {\mathsf{Sch}}^^{\operatorname{op}}\to{\mathsf{Sp}}\) in analogy to the functor \({\mathsf{K}}\), should look like a cohomology theory. A weak equivalence of schemes \(X \to Y\) would be expected to have the property that \(E(Y ) \to E(X)\) is a weak equivalence of spectra for every such \(E\).

Believing that this gives the ‘correct’ definition of weak equivalence reduces to believing that the whole homotopy theory of schemes is captured in the two basic facts that \(\mathrm{hocolim}~U_{-}\longrightarrow X\) and \(X \times I \to X\) are weak equivalences.

The surprising answer is **yes**. If we use this definition in the case of reasonable topological spaces (like CW-complexes) we do recover the usual notion of weak equivalence. It says that the homotopy-theory of topological spaces can indeed be ‘generated’ by these two fundamental properties.

The problem is that Zariski open covers don’t tell the whole story about the homotopy-type of schemes. The first indications of this date back to Grothendieck’s work on __Unsorted/etale cohomology__ : looking at only Zariski covers makes the spectrum of a field look contractible, whereas in real life they behave more like \(K(\pi, 1)\) (for example, they have nontrivial covering spaces).

We begin with the observation that algebraic topologists, despite what we are usually led to believe, are really not very interested in \({\mathsf{Top}}\). Of course there are certain subcategories, like the category of manifolds, in which they are highly interested – but the notion of ‘topological space’ is much too broad, admitting a host of pathological objects for which the machinery of algebraic topology is simply not intended.

On the other hand, categories such as topological manifolds end up being much too small for many purposes. The industrious homotopy-theorist finds himself wanting to glue manifolds together, quotient out by subspaces, divide out by group actions – in short, take various __colimits__ – all of which have the disadvantage of perhaps producing something which is no longer a manifold.

Gluing __manifolds__ together might lead one to study some category of cell complexes, but of course these categories end up not being closed under colimits either. Thus, what the algebraic topologist desires is a setting in which he can study the category of manifolds (or finite complexes, if he is daring), have all *small colimits* at his disposal, and yet does not have to worry about the pathology that goes along with objects as diverse as topological spaces.