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55
votes
10answers
10k views

Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
33
votes
2answers
2k views

What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
13
votes
2answers
2k views

What's about “quantum modular forms”?

Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling! Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...
11
votes
1answer
760 views

Rationalised K-theory of number fields

Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...
4
votes
1answer
706 views

Atiyah class for non-locally free sheaf

Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$. The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the ...
6
votes
3answers
1k views

The localisation long exact sequence in K-theory over an arbitrary base

If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory ... Kn+1(Dx) --> Kn(k) ...
9
votes
2answers
373 views

Genus of smooth varieties with small Chow group

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = ...
5
votes
6answers
2k views

Differences between reflexives and projectives modules

Let R be a normal noetherian domain. What is the difference between a finitely generated reflexive module and a finitely generated projective module? Can anybody recommend any references or make ...
4
votes
1answer
265 views

Adams graded parts of rational K-theory of a number field.

Let $F$ be a number field and $r_{1}$ and $r_{2}$ the numbers of real and pairs of complex embeddings respectively of $F$. Then Borel computed that for $n\geq 2$ $$ K_{n}(F)_{\mathbb{Q}}\simeq ...
2
votes
0answers
158 views

K-theory and completion

I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community. I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...