By Daniel Dugger
Read or Download A geometric introduction to K-theory [Lecture notes] PDF
Similar algebraic geometry books
This can be the 1st present quantity that collects lectures in this very important and quick constructing topic in arithmetic. The lectures are given via prime specialists within the box and the diversity of subject matters is stored as large as attainable through together with either the algebraic and the differential facets of noncommutative geometry in addition to contemporary functions to theoretical physics and quantity concept.
As somebody who heavily studied summary arithmetic and one whose father was once a bunch theorist earlier than getting into nuclear engineering, i've got constantly had an curiosity within the tough mathematical difficulties that brand new mathematicians are tackling. it sort of feels to me to be an fulfillment I by no means anticipated in my existence time to work out the 4 colour challenge and Fermat's final theorem either solved.
The class of algebraic surfaces is an tricky and engaging department of arithmetic, constructed over greater than a century and nonetheless an lively quarter of study this day. during this booklet, Professor Beauville supplies a lucid and concise account of the topic, expressed easily within the language of recent topology and sheaf conception, and available to any budding geometer.
The speculation of elliptic curves is extraordinary by way of its lengthy background and through the variety of the equipment which have been utilized in its examine. This publication treats the mathematics procedure in its sleek formula, by utilizing uncomplicated algebraic quantity concept and algebraic geometry. Following a quick dialogue of the required algebro-geometric effects, the publication proceeds with an exposition of the geometry and the formal workforce of elliptic curves, elliptic curves over finite fields, the complicated numbers, neighborhood fields, and worldwide fields.
Extra resources for A geometric introduction to K-theory [Lecture notes]
It suffices to prove that F (β, tu) = F (β, u) for any t ∈ S; for if u is another choice for u then 46 DANIEL DUGGER we would have F (β, u) = F (β, u u) = F (β, u ). 12, applied twice). Let us now write F (β) instead of F (β, u). The last thing that must be checked is that F (β ⊕ β ) = F (β) + F (β ), but this is obvious. So we have established the existence of ∂ : K1 (R) → K0 (R, S) having the desired properties. 17 (Localization sequence for K-theory). Let R be a commutative ring and S ⊆ R a multiplicative system.
A GEOMETRIC INTRODUCTION TO K-THEORY 39 If M is an n-multicomplex then let CM denote the cone on the identity map M → M . This is an (n + 1)-multicomplex, defined in the evident manner. This cone construction induces a group homomorphism K n−exct (R) → K (n+1)−exct (R). 10. The map K n−exct (R) → K (n+1)−exct (R) is an isomorphism, with inverse given by χ (M ) = (−1)j+1 j[Mj, ] where the symbols Mj, represent the various slices of M in any fixed direction. Proof. 2 almost verbatim, but where each Pi represents an n-exact multicomplex rather than an R-module.
This cone construction induces a group homomorphism K n−exct (R) → K (n+1)−exct (R). 10. The map K n−exct (R) → K (n+1)−exct (R) is an isomorphism, with inverse given by χ (M ) = (−1)j+1 j[Mj, ] where the symbols Mj, represent the various slices of M in any fixed direction. Proof. 2 almost verbatim, but where each Pi represents an n-exact multicomplex rather than an R-module. We have the sequence of isomorphisms K(R) → K exct (R) → K 2−exct (R) → · · · The composite map K(R) → K n−exct (R) sends [P ] to the n-dimensional cube consisting of P ’s and identity maps.
A geometric introduction to K-theory [Lecture notes] by Daniel Dugger