---
date: 2023-02-04 20:00
title: K Theory
aliases:
  - K Theory
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status: ❌ Not Started
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page_total: 100000
flashcard: 
created: 2023-02-04T20:00
updated: 2024-01-29T17:51
---

Last modified: `=this.file.mday`

# K Theory

- [x] What is Quillen's construction of $\K(A)$? ✅ 2023-02-04
	- [ ] $\pi_* \BGL_\infty(R)^+$
- [x] What is $\K_0(F)$ for $F$ a field? ✅ 2023-02-04
	- [ ] $\K_0(F) = \mods{F}^{\fg, \proj, \cong, \gp} = \ZZ_{\geq 0}^\gp = \ZZ$.
- [x] Describe $\K_1(A)$ for $A\in \Ring$. ✅ 2023-02-04
	- [ ] $\K_1(A) = \GL(A)^\ab = \GL(A)/E(A)$.
- [x] What is the $s\dash$cobordism theorem? ✅ 2023-02-04
	- [ ] If $W$ is an $h\dash$cobordism from $X$ to $Y$, when is $W$ a trivial cylinder?
	- [ ] Obstruction in a quotient of $\K_1(\ZZ[\pi_1 X])$.
- [ ] Describe how to construct $\BG$.
- [ ] What is the plus construction?
- [x] What is $\K(\FF_q)$? ✅ 2023-02-04
	- [ ] $$\K(\FF_q) = \ZZ + \bigoplus_{i \geq 1} C_{q^i-1}\cdot t^{2i-1}$$
- [x] Describe the state of computations for $\K(R)$. ✅ 2023-02-04
	- [ ] $R = \FF_q$ a finite field: known completely.
	- [ ] $R = F$ a field: known for $n=0, 1$.
	- [ ] $R = C_{p^k}$ for $k \geq 2$ currently unknown. 
	- [ ] $R = \ZZ$ not completely known.
	- [ ] $R = (\ZZ[x]/\gens{x^n}, \gens{x})$ partially known using trace methods: $\ZZ^{m-1}$ in odd degrees, order $\approx (mi)!(i!)^{m-2}$ in even degrees $2i$.
	- [ ] $R = \ZZ[x,y]/\gens{xy}, F[x,y]/\gens{xy}$ for $F$ characteristc $p$ well-understood.
- [x] Give example applications of computing $\K(R)$. ✅ 2023-02-04
	- [ ] Vandiver's conjecture: for $p$ prime and $K \leq \QQ(\zeta_p)$  the maximal real subfield, $p\nmid \cl(K)$. Equivalent to $\K_{4n}(\ZZ) = 0$ for all $n\in \ZZ_{\gt 0}$.
- [ ] How is motivic cohomology defined?
- [x] What is the Atiyah-Hirzebruch spectral sequence? ✅ 2023-02-04
	- [ ] $H^*_\sing \abuts \K^\Top$.
- [x] What is the cyclic bar construction? ✅ 2023-02-04
	- [ ] $B_n^{\mathrm{cyc}}(A) \da A\tensorpower{}{n+1}$ with 
	- [ ] Differentials: collapsing $\del_i(\vector a) = a_0\tensor \cdots \tensor a_i a_{i+1}\tensor\cdots\tensor a_n$ for $0\leq i\leq n-1$ and cyclcing $\del_i(\vector a) = a_n a_0\tensor\cdots \tensor a_{n-1}$ for $i=n$.
	  - [ ] Form the full differential as $\del = \sum_i (-1)^i \del_i$.
	- [ ] Degeneracy maps $s_i$ insert the unit into the $i$th coordinate.
	- [ ] $n$th level carries an action of $C_{n+1}$ where the generator acts by $\vector a \mapsto a_n \tensor a_0 \tensor \cdots \tensor a_{n-1}$.
- [x] Define $\HoH_n(A)$. ✅ 2023-02-04
	- [ ] The homology of the cyclic bar complex, $H_n(C_\bullet(A))$, or by Dold-Kan, $\pi_n \realize{B_\bullet^{\mathrm{cyc}}(A)}$, the geometric realization of a simplicial object.
- [ ] What is the Dold-Kan correspondence?
- [x] Why are cyclic objects $X$ important? ✅ 2023-02-04
	- [ ] $S^1\actson \realize{X}$.
- [x] What is the Dennis trace? ✅ 2023-02-04
	- [ ] $\K_q(A) \to \HoH_q(A)$.
- [x] Why is Hocshild homology an approximation of K-theory? ✅ 2023-02-04
	- [ ] The Dennis trace factors as $\K_q(A) \to HC^-_q(A)\to \HoH_q(A)$.
	- [ ] For $I\normal A$ nilpotent, $\K_q(A, I) \tensor \QQ \iso HC^i_q(A\tensor \QQ, I\tensor \QQ)$.
- [x] What does $\THoH(A)$ mean? ✅ 2023-02-04
	- [ ] $\THoH(HA)$ for $HA$ the Eilenberg-Maclane spectrum.
	- [ ] Idea of construction: replace $\ZZ$ with $\SS$ and $\tensor_\ZZ$ with $\smashprod$ to form acyclic bar complex.
- [ ] What is a genuine $S^1\dash$spectrum?
- [ ] What is a cyclotomic structure?
- [ ] What is the cyclotomic trace?
	- [ ] $\K(A) \to \TC(A)$ where $\TC(A; p) = \mathrm{ho}\cocolim_{R, F} \THoH(A)^{C_{p^n}}$ as the homotopy limit of a tower: ![](attachments/2023-02-04-holim-tower.png)
- [x] Why is $\TC(A)$ a good approximation of $\K(A)$? ✅ 2023-02-04
	- [ ] For $I\normal A$ nilpotent, $\K_q(A; I) \iso \TC_q(A; I)$.
- [ ] What is the Quillen-Lichtenbaum conjecture?
- [ ] What are cyclotomic spectra?
- [ ] Concretely describe $p\dash$completion for $\zmod$.
- [ ] What is crystalline cohomology?
- [ ] What is $p\dash$adic Hodge theory?