Overview
We will discuss a number of foundational results in algebraic K-theory arising from the additivity theorem. This will include localisation sequences as well as devissage-type statements like the resolution theorem, the theorem of the heart and the Gillet-Waldhausen theorem. On the way, we will develop the basics of exact and (pre)stable ∞-categories, localisations of ∞-categories, weight structures and t-structures.
Prerequisites: The course will assume some familiarity with ∞-categories and the ∞-category of spectra. The main examples will arise in the form of categories of parametrised spectra and derived categories, even though we will also give a self-contained definition of derived categories.
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Literature
A few pointers to some relevant literature: Waldhausen's seminal paper, which also officially introduced the S.-construction, is
- F. Waldhausen. Algebraic K-theory of spaces. Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 318-419 (1985).
A more modern and very concise source for the algebraic K-theory of stable categories is
- F. Hebestreit, A. Lachmann, W. Steimle. The localisation theorem for the algebraic K-theory of stable ∞-categories. arXiv:2205.06104
which in particular builds on the following articles dealing with the universal property of algebraic K-theory:
- C. Barwick. On the algebraic K-theory of higher categories. J. Topol. 9, No. 1, 245-347 (2016).
- A. Blumberg, D. Gepner, G. Tabuada. A universal characterization of higher algebraic K-theory. Geom. Topol. 17, No. 2, 733-838 (2013).
The things we need to know about (pre)stable categories will mostly be taken from chapter 1 respectively appendix C in
That the homotopy category of chain complexes is stable is a consequence of results in
- A. Blumberg, M. Mandell. Algebraic K-theory and abstract homotopy theory. Adv. Math. 226, No. 4, 3760-3812 (2011).
- M. Weiss. Hammock localization in Waldhausen categories. J. Pure Appl. Algebra 138, No. 2, 185-195 (1999).
But the proof we are going to give is taken from chapter 7.2 in
- D.-C. Cisinski. Higher categories and homotopical algebra. Cambridge Studies in Advanced Mathematics 180 (2019).
Weight structures were originally introduced by Bondarko in the setting of triangulated categories.
A concise reference for weight structures on stable ∞-categories can be found in section 3 of
- F. Hebestreit, W. Steimle. Stable moduli spaces of hermitian forms. arXiv:2103.13911
The equivalence between stable ∞-categories with bounded weight structures and weakly idempotent complete additive ∞-categories was proven in
- V. Sosnilo. Theorem of the heart in negative K-theory for weight structures. Doc. Math. 24, 2137-2158 (2019).
The main sources about exact ∞-categories are
- C. Barwick. On exact ∞-categories and the theorem of the heart. Compos. Math. 151, No. 11, 2160-2186 (2015).
- J. Klemenc. The stable hull of an exact ∞-category. Homology Homotopy Appl. 24, No. 2, 195-220 (2022).
Even though it is only concerned with ordinary exact categories,
- T. Bühler. Exact categories. Expo. Math. 28, No. 1, 1-69 (2010)
can be a very helpful reference. In particular, most of the diagram lemmas can be lifted to exact ∞-categories with only minimal modifications.
Later sections also contain a discussion of derived categories (in the language of triangulated categories).
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