Lecturer: Fabian Hebestreit
Office: 3.023, not that it matters anymore
eMail: f.hebestreit at math.uni-bonn.de
Lectures:
Slot 1: tuesdays, noon - 2pm
Slot 2: thursdays, noon - 2pm
Location: Zoom, meeting ID: 938 9702 5919

Algebraic and Hermitian K-Theory

News

Updated script with the construction of non-connective K-spectra, some outlook on the K-groups of the integers, and the theorem of Land-Tamme. I'll still add a brief discussion of the so-called fundamental theorem (which computes K-groups of Laurent-polynomial rings) and Madsen-Weiss in the coming days, for whoever is still coming by here sometimes.
Chapter 2 is now also complete, though the end is still a bit rough (weird page breaks and some missing numbering). Starting roughly around page 100 I have tried to give an overview of the relevant material of Higher Algebra, sometimes with simpler proofs in the special cases that we need, sometimes only by reference. Please treat it as a guide towards reading that book, rather than requisite material for the lecture.
I'll leave the following warnings from last year here for a while: I corrected a mistake in the recognition of operads as symmetric monoidal categories. Thanks to Bastiaan and Sil for sticking to their (equally wrong) point about two-fold tensor products, until we found this error of mine!
I also corrected a small error in the definition of the Day convolution structure (beware that this error originates in Nikolaus' paper, on which the lecture was built).

Zoom

If you want to participate in the lecture, just send me an email and I will give you the password.

Exams

The first batch of exams will take place in the week of February 15th - 19th. Let's arrange further details after Christmas.
The second batch will happen in the week of March 22nd - 26th.

Overview

In the lecture we will go through some of the basics of algebraic K-theory, largely in the language of higher categories. Some familiarity with that language will therefore be required, though not much technical detail. The algebraic K-theory of a ring R was defined by Quillen as the group completion of the symmetric monoidal groupoid of finitely generated projective R-modules. This group completion happens in the world of E_∞-spaces a.k.a symmetric monoidal ∞-groupoids a.k.a Picard anima, which is how higher category theory comes into play. The plan is to go through some of:

  • Symmetric monoidal ∞-categories and E_∞-spaces
  • Stable ∞-categories and spectra
  • The recognition principle and the group-completion theorem
  • The K-Theory of finite fields
  • The K-Theory of stable ∞-categories and the theorem of the heart
  • K-theory as the universal additive invariant
  • Hermitian K-theory and L-theory
This is a large amount of material and the plan is not to cover all of it in detail (or maybe at all, depending on how the semester progresses); for example, I will largely treat the requisite higher category theory that facilitates the discussion only in survey form. Beside some knowledge of higher categories, good knowledge of homology (both in the world of topology and of algebra) will be required for much of the course.

Notes

Ferdinand is also going through the troubles of texing the lectures. His notes can be found here, I will not generally check them for correctness, but what I have checked is really nice (and of course this also does not mean that mine are more correct).

Recommended Literature

To be updated as we go along.

Large parts of the lecture will be taken straight out of

  • Lurie: Higher Algebra (available here)
For an account of the K-theory of stable categories I will shamelessly plug

  • Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus, Steimle: Hermitian K-theory of stable ∞-categories: I, II, III (available from my website)
and for the classical computation for finite fields nothing beats

  • Quillen: On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field
For general education in K-Theory, there is Weibel's K-book (just google it!) and as references for basic higher category theory:

  • Cisinski: Higher categories and homotopical algebra (available here)
  • Lurie: Higher Topos theory (available here)
  • Lurie: Kerodon (Wiki, available here)
  • My scripts from last year (available from my webpage)
Here are some more specific references:

  • Spaltenstein: Resolutions of unbounded complexes (for the verification that D(R) is a Bousfield localisation of K(R))
  • Rezk: A model for the homotopy theory of homotopy theories (the original source of complete Segal spaces)
  • Joyal, Tierney: Quasi-categories vs Segal spaces (for the original proof that complete Segal spaces and quasi-categories give the same ∞-category)
  • Lurie: (∞,2)-categories and Goodwillie calculus (for a proof, which embeds this into the greater context of (∞,n)-categories)
  • Nikolaus: Stable ∞-operads and the multiplicative Yoneda lemma (for a thorough discussion of Day convolution and stable operads)
And finally the classic references in which the subject of higher K-theory was born:

  • Quillen: Higher algbraic K-Theory I
  • Waldhausen: Algebraic K-Theory of spaces
  • Thomas and Trobaugh: Higher algebraic K-Theory of Schemes and of Derived Categories