# A geometric introduction to K-theory [Lecture notes] by Daniel Dugger By Daniel Dugger

Best algebraic geometry books

Introduction to modern number theory : fundamental problems, ideas and theories

This variation has been known as ‘startlingly up-to-date’, and during this corrected moment printing you'll be convinced that it’s much more contemporaneous. It surveys from a unified viewpoint either the trendy kingdom and the developments of continuous improvement in a number of branches of quantity idea. Illuminated by means of simple difficulties, the vital principles of recent theories are laid naked.

Singularity Theory I

From the studies of the 1st printing of this booklet, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My common effect is of a very great publication, with a well-balanced bibliography, prompt! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors supply right here an up to the moment consultant to the subject and its major functions, together with a couple of new effects.

An introduction to ergodic theory

This article presents an creation to ergodic thought compatible for readers realizing easy degree concept. The mathematical must haves are summarized in bankruptcy zero. it truly is was hoping the reader should be able to take on examine papers after examining the ebook. the 1st a part of the textual content is anxious with measure-preserving differences of likelihood areas; recurrence houses, blending homes, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy conception are mentioned.

Additional resources for A geometric introduction to K-theory [Lecture notes]

Example text

The map GL(R)ab → K1 (R) is an isomorphism. It will be convenient to prove this at the same time that we give other descriptions for K1 (R). In particular, we make the following definitions: (1) K1f r (R) is the group defined similarly to K1 (R) but changing all occurrences of ‘projective’ to ‘free’. 5). The “sp” stands for “split”. (3) K1sp,f r (R) is the group defined by making both the changes indicated in (1) and (2). 7) GL(R)ab colimP Aut(P )ab y G G K sp (R) 1 y G G K1 (R) y colimn GLn (R)ab G G K sp,f r (R) 1 G G K f r (R).

We can write r = uπ k for some unit u ∈ R and n ≥ 0, in which case R/rR ∼ = R/π k R and so (R/rR) = k. It follows that the composite ∂ ∼ = ∼ = F ∗ −→ K0 (R, S) −→ G(M | S −1 M = 0) −→ Z is just the usual π-adic valuation on F ∗ . The following example generalizes the previous one, but is a bit more interesting. 19. Let D be a Dedekind domain—a regular ring of dimension one. In such a ring all nonzero primes are maximal ideals. Let S = D − {0} and let F = S −1 D be the quotient field. Our localization sequence looks like K1 (D) → F ∗ → K0 (D, S) → K0 (D) → Z.

Proof. Use the following short exact sequence of maps: 0 GP f 0 g idQ ⊕βα α  GQ G Q⊕P f  G Q⊕W g GQ G0 β  GW G 0, A GEOMETRIC INTRODUCTION TO K-THEORY 33 where f (x) = (α(x), x), f (y) = (y, β(y)), g(a, b) = a − α(b), and g (c, d) = β(c) − d. This gives that βα id β α [Q −→ Q] + [P −→ W ] = [P −→ Q] + [Q −→ W ], but of course the first term on the left is zero in K(R, S). Note that there is an evident map K(R, S) → K(R) that sends a class [P ] in K(R, S) to the similarly-named (but different) class [P ] in K(R); in colloquial terms, the map simply ‘forgets’ that a complex P is S-exact.