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Algebraic K-theory and localizing invariants SemesterWiSe 2022 / 23 Lecturer Denis-Charles Cisinski Type of course (Veranstaltungsart) Vorlesung German title K-Theorie und lokalisierende Invarianten Contents K-Theory is the abstraction of enumerative invariants from linear algebra such as the rank of (locally) free modules, determinants of endomorphisms of these, or the length of finite modules. It has an homotopy theoretic flavour: for instance, the notion of rank leads to the notion of Euler characteristic of a bounded complex. This has many applications in Topology, Algebraic Geometry, and in Representation Theory, since it provides rather powerful tools to define and compute enumerative invariants (e.g. the Lefschetz fixed point formula in topology, zeta functions of algebraic varieties over finite fields, intersection multiplicities in Chow groups). On the other hand K-Theory is a kind of homology theory that can be applied to an extremely broad class of objects: at the very least any stable category. As such this is also a very efficient device to capture the structure of these objects (e.g. understanding the properties of Verdier quotients). This is a versatile tool with a great power of expression which is notoriously very hard to compute, but that has strong and fruitful interactions with other homology theories (e.g. topological Hichschild homology and de Rham cohomology, higher Chow groups). The goal of this lecture will be to introduce the definition of algebraic K-theory, possibly with some variants (non-connective K-theory, homotopy invariant K-theory) and to establish its basic properties (Additivity Theorem, Localization Theorem, Dévissage), with full proofs. We will mainly focus on examples and applications related to Algebraic Geometry, illustrating the general theory by proving fundamental properties of K-theory of schemes (mainly, Poincaré duality, homotopy incariance, descent theorems), revisiting classical results of Quillen and Thomason. We will then explain how to determine algebraic K-theory through several universal properties. In particular, we will revisit the work of Andrew J. Blumberg, David Gepner, Goncalo Tabuada, which leads to a theory of non-commutative motives, following insights of Kontsevich. This will be the starting point to compare K-theory with other classical localizing invariants, in particular (topological) Hochschild homology. If times permits, we will use these tools to revisit the Grothendieck-Riemann-Roch theorem. Recommended previous knowledge We will speak freely the language of higher category theory (but we will do our best to make the lecture accessible to anyone having faith in the fact such a thing exists and willing to accept the basics as a black box) - to be more precise, we will work with the model of quasi-categories. Many examples will come from algebraic geometry so that it is recommended to know basic Algebraic Geometry (in particular, the very notion of scheme, and the (derived) category of sheaves with quasi-coherent cohomology). Time/Date Mo 16-18, Fri 16-18 Location Mo M 102, Fri M 103 and online Course homepage https://cisinski.app.uni-regensburg.de/lehre.html (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BV, MV, MArGeo, MGAGeo ECTS 9 |