Talks: Friday 1:30-3:00 PM GMT-5, Room 4N30 , David Rittenhouse Laboratory, 209 S 33rd St, Philadelphia, PA 19104.
Problem sessions: 2:15-3:00 PM during the lecture time.
This is a reading course of algebraic K theory at Penn, Fall 2023. Advisor: Prof. Daniel Krashen.
Here is the syllabus of this course.
Problem sessions: 2:15-3:00 PM during the lecture time.
This is a reading course of algebraic K theory at Penn, Fall 2023. Advisor: Prof. Daniel Krashen.
Here is the syllabus of this course.
Eastern Standard Time (EST)
Notes
- Week 1 (Sep 8): Lecturer: Daebeom Choi \( K_0 \) of rings, Serre-Swan (category of finite rank vector bundle of Hausdorff compact space is equivalent to the category of finitely generated projective modules over \( C^{\infty}(M) \) ), computation of special rings, Morita invariance of \( K_0 \), Dedekind domains, class groups, \( K_0 \) of Dedekind domains.
- Week 2 (Sep 15): Lecturer: Yaojie Hu Projective modules, pushouts and pullbacks, construction of projectives, elementary matrices, Whitehead groups \( GL(R)/E(R) \), definition of \( K_1 \), Morita invariance of \( K_1 \), \( K_1 \) of local rings and division rings, exact sequences associated with pullbacks and pushouts.
- Week 3 (Sep 22): Lecturer: Fangji Liu Relations in \( E(R) \), perfect ring, Steinberg groups, definition of \( K_2 \), (universal) central extensions, example of central extensions, detection theorem (\( G \) perfect iff G admits a universal central extension), relations to group homology, properties of \( K_2 \)
- Week 4 (Sep 29): Lecturer: Chayansudha Biswas Review of \( K_2 \), Steinberg Symbol and related identities, examples of Steinberg symbol: (\( K_2 = \mathbb{Z}/2 \) generated by \( \{ -1, -1\} \), coordinates replacement matrices, computations in Steinberg groups, subgroups \( W = \left \langle w_{ij}(u) \right \rangle \) and \( H = \left \langle h_{ij}(u) = w_{ij}(u) w_{ij}(-1) \right \rangle \) of Steinberg group, decomposition of codomain of \( W \to GL(n,R) \) into permutation and diagonals, special case of Weyl groups, subgroups generated by Steinberg symbol.
- Week 5 (Oct 6): Lecturer: Marc Muhleisen The long exact sequence of \( A, a \) in \( K_0, K_1 \), Mayer-Victoris.
- Week 6 (Oct 13): FALL BREAK!
- Week 7 (Oct 20): Lecturer: Jinghui Yang Acyclic spaces and maps, properties of these objects, classifying space, Quillen's +-construction, the definition of higher algebraic K groups \( K_n(R) = \pi_n (BGL(R))^+ \) (\( n \geq 1 \)), K-spaces \( K(R) = K_0(R) \times BGL(R)^+ \), concrete construction of +-constructions via attaching 2- & 3-cells, verification of new definition coinciding the traditional definition of \( K_0, K_1, K_2 \), Barratt-Priddy-Quillen-Segal theorem, +-construction is infinite loop space, May's recognition theorem on K-spaces, computation of stable stem \( \pi_1^s, \pi_2^s \).
- Week 8 (Oct 27): Lecturer: Marc Muhleisen Recapture of classical \( K_2 \), Steinberg groups, Steinberg symbols, valuation, valuation rings, properties of valuation rings (local, criterion), Matsumoto's theorem \( K_2(F) = F^{\times} \otimes_{\mathbb{Z}} F^{\times} / \left \langle x \otimes (1-x) \right \rangle_x \), Steinberg symbols on fields valued in abelian groups, formulas of Steinberg symbols with discrete valuation, calculation of \( K_2 \mathbb{Q} \cong \{ \pm \} \oplus \mathbb{F}_3 \oplus \mathbb{F}_5 \oplus \cdots \), quadratic reciprocity, image of \( K_2 \) into \( \{ \pm \} \), relation to Steinberg symbol, Weil theorem, proof of Matsumoto's theorem.
- Week 9 (Nov 3): Lecturer: Jinghui Yang Prerequisite: simplicial objects, nerves, geometric realizations, homotopy theory of categories. See this notes. (Please let me know if there are any mistakes or typos)
- Week 10 (Nov 10): Lecturer: Kartik Tandon Waldhausen categories, K-theory of Waldhausen S-construction, "Quillen Q = Waldhausen S-construction", K-theory of schemes, stable \( \infty \)-categories, derived categories (of schemes), t-structures, \( D(X)^{\heartsuit} = \text{QCoh}(X) \), perfect complexes, compact objects, \( \text{Idem}(\mathcal{C}), \text{Ind}(\mathcal{C}) \), categories of (small, idempotent-complete, presentable) stable \( \infty \)-categories, Verdier quotients, Karoubi sequences, Neeman-Thomason localization, additivity theorem, universality theorem, Grothendieck topology, Zariski descent, Nisnevich descent, Beilinson theorem, \( K(\mathbb{P}_R^2) = K(R) \oplus K(R) \).
- Week 11 (Nov 17): Lecturer: Daebeom Choi Review of exact categories, Q-constructions, Quillen's theorem A & B, and fundamental theorems (Resolution, Localization, Dévissage). Corollary on \( K_n(\mathsf{Mod}_{A}^{f.g.}[f^{\infty}]) \), long exact sequences for regular Noetherian \( A \) ( \( \cdots \to K_n(A/f) \to K_n(A) \to K_n(A_f) \to \cdots \) ), the case of the Dedekind domains and their fraction fields. Serre subcategories (or thick subcategories), proof sketch of Dévissage (the rest two in notes), K-theory of Dedekind schemes, coherent sheaves of support in codimension \(p\) ( \( \mathsf{Coh}^p(X) \) ). filtration of \( \mathsf{Coh}(X) \) w.r.t. \( \mathsf{Coh}^p(X) \), Brown-Gersten-Quillen (BGQ) spectral sequences, Chow groups, Bloch's formula.
- Week 12 (Nov 24): THANKSGIVING BREAK!
- Week 13 (Dec 1): Lecturer: Fangji Liu Representation rings \( R(G) \), Adams operations \( \Phi^k \), construction of Adams operations, non-commutative diagram of \( \Phi^k \) and \( \beta \) (Bott periodicity), fiber sequence \( F \Phi^q \to BU \xrightarrow{\Phi^q -1} BU \), characters of representations, Adams operations on representation (ring), Brauer lifting, construction of map \( g^+: BGL \mathbb{F}_q^+ \to BU \), lifting of \( F \Phi^q \), homotopy equivalence of \( BGL \mathbb{F}_q^+ \to F \Phi^q \), Quillen's calculus of K-theory of finite fields, K-theory of finite coefficients, Suslin's rigidity theorem, K-theory of algebraic closed fields \( F \) (\( \text{char} \, F = p >0 \) or 0 ) in finite coefficients, torsions of \( K_{2i-1}(F) \), result of \( K_{\ast}(\mathbb{Z}) \), \( e \)-invariant, Vandiver conjecture.
- Week 14 (Dec 12): Lecturer: Yaojie Hu Hochschild homology, cyclic operator, (cyclic) bar construction, \( HH_n(A, M) \cong \text{Tor}^{A^e}(M, A) \), Hochschild-Kostant-Rosenberg (\( HH_n(A) \cong \Omega^n_{A/k} \)), (negative) cyclic homology and periodic homology, Connes complex, Killing contractible complex, Connes operator \( B=(1-t)sN \), Connes periodicity exact sequences, relation between \( HC, HC^-, HP \) and Kahler differentials, (algebraic) connections, (algebraic) de Rham cohomology theory, Levi-Civita connections, Chern characters, Dennis trace map \( K_n(A) \to HC^-(A) \).
- END OF SEMESTER!
See Notes.
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See Notes. Note that I add some extra content in this note of my own flavor.
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Exact categories, resolution w.r.t. subcategory, Resolution theorem, Quillen's Q-category of exact categories, correspondence between isomorphism, Q-construction of K-theory, "+=Q", \( \infty \)-categories (quasi-category model), additive \( \infty \)-categories, stable \( \infty \)-categories, exact \( \infty \)-categories, Quillen's Q-category of exact \( \infty \)-categories, classical Q-construction equivalent to \( \infty \)-categorical Q-construction, thick subcategories, Localization Theorem, Dévissage.
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Extras
- A non-trivial h-cobordism: Akbulut cork.
- About \( SK_1 \): see this post and this post, and this note.
- \( A_{n}/A_{\infty}/E_{n}/E_{\infty} \)-operad, recognition principle, approximation theorem: see this 2023 iWoAT summer school and the book \( E_{\infty} \) Ring Spaces and \( E_{\infty} \) Ring Spectra by Peter May.
References
- [Milnor] Introduction to Algebraic K-Theory We mainly follow this book.
- [Rosenberg] Algebraic K-Theory and Its Applications . This is a very good resource for the course!
- [Quillen] Higher algebraic K-theory: I . This is a good resource for algebraic topologists.
- [Adams] Infinite Loop Spaces . It's a very classical and great resource.
- [Weibel] The K-book: an introduction to algebraic K-theory . Another main reference for (especially) higher algebraic K-theory.
Useful Links
- Math 7020: Algebraic K-theory and Algebraic Cycles (Fall 2022). Taught by Prof. Daniel Krashen, University of Pennsylvania.
