This is an overview on Serre-Swan theorem and some ideas on the construction of K-groups for a Banach category. Serre-Swan theorem establishes equivalences between the categories of topological vector bundles over a compact Hausdorff space , the category of finitely generated projective -modules and the categories of algebraic vector bundles of finite rank over
the affine scheme . This theorem connects different objects of interest in K-theory.
It also introduces some ideas on the construction of K-groups for a Banach category and
in particular for compact topological spaces and Banach algebras.
A simple visual editor for creating commutative diagrams.
Announcing quiver: a new commutative diagram editor for the web: https://varkor.github.io/blog/2020/11/25/announcing-quiver.html
There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.
The constructions look like these (see the links above for details):
But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.
The connected components [Background]
For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)
Two examples of (co)limits as (co)equalisers
Pdf here: Two examples of (co)limits as (co)equalisers
This is about an exercise in [Bass]:
Exercise 19.5. Prove that if is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in is not compact.
Proof. Choose an orthonormal basis , then . This means the sequence is not Cauchy hence has no convergent subsequence.
For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.
[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.
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.
Here is a MO post asking a question that I’ve had in mind for a while: Why higher category theory?
Having studied some elementary topos before, I have been interested in higher topos since I attended a summer school lecture last year. Besides formal generalisations, I expect to see applications which provide new results or meaningful insights. Though the meaning of “new” and “useful” very much diverse between different mathematical cultures.
Here are some important applications of higher categories in K-theory (added on Mar 05, 2020), suggested by my supervisor Schlichting.
Dustin-Mathew-Morrow, Algebraic K-theory and descent for blow-ups
Nikolaus-Scholze, On topological cyclic homology
Clausen-Mathew-Morrow, K-theory and topological cyclic homology of henselian pairs
Antieau-Gepner-Heller, K-theoretic obstructions to bounded t-structures
Blumberg-Gepner-Tabuada, A universal characterization of higher algebraic K-theory
Barwick, On exact infinity-categories and the Theorem of the Heart
This is my work on the six exercises Exercise 13.3.D-13.3.I in Section 13.3.3 of Vakil’s notes. We look at a useful characterisation of quasicoherent sheaves in terms of distinguished inclusions and prove some properties in reasonable circumstances (quasicompact and quasiseparated).
Here is the pdf file: Characterisation of quasicoherent sheaves