K-Theory

Motivations

The Lichtenbaum-Quillen conjectures related to etale cohomology, Kummer-Vandiver conjecture, relations to THH and TC.

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Notes

Algebraic K is analogous to ${\operatorname{KU}}$, complex topological K theory, or ${\operatorname{KO}}$, real topological K-theory.
- The Grothendieck-Witt group is analogous to ${\operatorname{KO}}$?

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Construction

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See Infinite loop space machine. attachments/Pasted%20image%2020220415125059.png

Categorical K Theory

Defining ${\mathsf{K}}_0$ for ${}_{R}{\mathsf{Mod}} $: Pasted image 20211103190932.png
Where it shows up naturally in algebraic topology: the Wall finiteness obstruction. The finiteness obstruction $w(X)$ is zero if and only if $X$ has the homotopy type of a finite CW complex. Pasted image 20211103191051.png
${\mathsf{K}}_1$ shows up in defining Whitehead torsion: Pasted image 20211103191524.png

Algebraic K Theory

It’s like a homology theory on $\mathsf{CRing}$.
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See Borel regulator, Lichtenbaum-Quillen conjectures, Zeta function.

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K(Z)

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In toplogy

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Examples

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Complex K Theory

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For stacks

See stacks MOC: attachments/Pasted%20image%2020220319213903.png

Misc

$X(R) \coloneqq( {}_{R}{\mathsf{Mod}} ^{\mathrm{free}})^\cong { \, \xrightarrow{\sim}\, }{\textstyle\coprod}_n \mathbf{B}\mkern-3mu \operatorname{GL} _n(R)$

Gaps

Build a simplicial set:

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Quillen’s construction

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Misc

See syntomic cohomology: attachments/Pasted%20image%2020220515001557.png

🗓️ Timeline

Prismatic cohomology

Link to K-theory comes from eigenspaces somehow.

The category $\mathsf{Prism}$ doesn’t have a final object, so has interesting cohomology. Relates to the algebraic K theory of ${\mathbf{Z}}_p$?

algebraic K theory is hard, using Topological Hochschild homology somehow makes computations easier.

The Relationship Between THH and K-theory
2021-06-06

Roughly twice as hard as computing K-theory with ku! (Wilson, Adams, Margalis)

This says either the Hilbert scheme or K-theory is hard.
2021-05-04

Allows detecting classes in $Z^2(X)$ using K-theoretic methods.
2021-04-25

How can we construct this using modern groupoid yoga? Take the category ${}_{G}{\mathsf{Mod}}$, somehow restrict to just irreducible representations. Maybe there’s a better thing to do here though, like “ignoring” reducibles the same way John Carlson “ignored” projectives. But okay, anyway, take that category. Take its nerve and then the geometric realization and then $\pi_0$ or something? And then take the free ${\mathbf{Z}}{\hbox{-}}$module. I definitely need to ask some homotopy theorists how this construction goes for usual K-theory in modern terms. So like… \begin{align*} {\mathbf{Z}}\left[ \pi_0 {\left\lvert { N \mathsf{C} } \right\rvert} \right] .\end{align*} The $\pi_0$ should be taking isomorphism classes somehow, but maybe this only works for groupoids? But okay, whatever, I just need a functor that takes categories into spaces where two objects end up in the same path component iff they’re isomorphic in $\mathsf{C}$. So maybe this needs to be something more simplicial set.