Starting with some motivations and brief expositions on algebraic K theory, I’ll introduce some early important computations of algebraic K-theory, including computations of K theory of finite fields and of rings of integers for which I will briefly outline the proofs. Then we’ll move on to K-theory with finite coefficients of separably closed fields. With the motivation of recovering some information of K-theory of an arbitrary field from its separable closure, we introduce a few versions of the Lichtenbaum-Quillen conjectures as descent spectral sequences of \’etale Cohomology groups. If time permits, I’ll mention relation to motivic Cohomology that a key tool is some “motivic-to-K-theory” spectral sequence.
These are notes based on my talk on Oct 01, 2021 in UIUC Graduate Homotopy Seminar. The main references are  and . The outline of the proof of Quillen’s K-theory of finite fields has been moved to Appendix A.
Here are the notes, updated on 10/30/2021. I thank Prof Grayson for comments and pointing out some typos.
Updates (10/30/2021): Fixing a few typos, adding reference to the claim about the fixed point spectrum for a Galois extension .
 Mitchell, Stephen A. “On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint.” Algebraic topology and its applications. Springer, New York, NY, 1994. 163-240.
 Weibel, Charles A. The K-book: An introduction to algebraic K-theory. Vol. 145. Providence, RI: American Mathematical Society, 2013.
This is an outline of Quillen’s proof for the calculation of K-theory of finite fields, originally done by Quillen in , see also  for a slightly different presentations with more background materials included.
Here are the full post:
 Quillen, Daniel. “On the cohomology and K-theory of the general linear groups over a finite field.” Annals of Mathematics 96.3 (1972): 552-586.
 Mitchell, S. Notes on K theory of nite elds. Available online:
This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here: 2018-2019 Ring Theory. Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.
Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].
See here for the full post: Application of Krull’s Principal Ideal Theorem
Please also let me know if you find any errors or have suggestions on any of my posts.