Algebraic K-theory
Summer 2024
Algebraic K-theory
Summer 2024
Algebraic K-theory is a problem child. It is very difficult to work with and very difficult to define; but, like all good problem children, when it shines, it really shines. At its core, algebraic K-theory can be used to relate arithmetic, algebraic, and topological information to make meaningful, powerful, and deep statements. Some examples of its uses are:
The Grothendieck-Riemann-Roch theorem, essentially what Grothendieck invented K-theory for, generalizes the classical Riemann-Roch theorem to quasi-projective schemes over some field, and it is best told by constructing a map from the 0-th K-group to the Chow ring.
The Milnor conjecture, which gives an isomorphism between the Milnor K-theory of a field (an arithmetic and computable variant of algebraic K-theory) and Galois cohomology.
Motivic homotopy theory, which uses algebraic K-theory as an analogue of complex K-theory to study the category of motivic spectra.
Trace methods, which is the study of various maps generalizing the trace of a matrix as a way to understand Hochschild and cyclic homology, as well as their topological variants.
This seminar will be an investigation in algebraic K-theory, both in its construction and its applications. As there are multiple different ways to come at this subject and motivation for learning it, in addition to summer traveling, I propose instead to make this a more relaxed seminar. There is a gigantic reference book called The K-book by Weibel. It is encyclopedic, and covers more than any one mortal should learn. My proposition, then, is to have this seminar be in the style of a reading seminar with write-ups where we meet in whatever capacity we can whenever we can. A good goal would be to go through chapters 2, 3, and 4 of the K-Book before going into selected topics.
Resources: