# Tagged Questions

**18**

votes

**5**answers

644 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...

**33**

votes

**2**answers

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 ...

**1**

vote

**0**answers

134 views

### $K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...

**14**

votes

**2**answers

789 views

### Who first noticed that the Hilbert symbol is a Steinberg symbol ?

Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula
$$
\prod_v(a,b)_v=1
$$
for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ ...

**6**

votes

**1**answer

237 views

### Projective modules over non-rational group rings

Let $G$ be a finite group. We know that the $K$-group $K_0(QG)$ of the rational group ring $QG$ is a free abelian group generated by the irreducible representations of $G$ over $Q$. Now let $R$ be a ...

**13**

votes

**1**answer

367 views

### Beilinson's formula for the product of two modular curves

In his cellebrated 1984 paper "Higher regulators and values of L-functions", Beilinson proved (among many other exciting things) that the value at the non-critical point $s=2$ of the Rankin L-function ...

**5**

votes

**3**answers

557 views

### Infinite dimensional central simple algebras

When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization).
But what happens to the ...

**4**

votes

**2**answers

771 views

### Survey of Algebraic K-Theory Since 1980?

I just came across Charles Weibel's Development of Algebraic K-Theory until 1980, and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? ...

**8**

votes

**3**answers

1k views

### A question on K_1 of an elliptic curve

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , ...

**2**

votes

**1**answer

329 views

### Zero-cycles on an arithmetic surface

Could anyone give a reference for the following statement, which I believe is true.
"Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ ...

**1**

vote

**0**answers

174 views

### How to get countably many generators for $K_{j}^{G}(\beta G)$ ??

Hey
I am trying to find out how the Baum-Connes conjecture works over $GL(1)$ over local fiels.
I am just wondering if anybody knows how to get a countable many generators for in the L.H.S of the ...

**9**

votes

**1**answer

525 views

### Numerical evidence of Beilinson's conjecture in local fields and function fields

The famous Beilinson's conjecture predicts a relationship between the regulator map in $K$-theory and special value of $L$-function generalizing the Dirichlet's theorem in number theory. Please see ...

**14**

votes

**1**answer

823 views

### Dirichlet's regulator vs Beilinson's regulator

Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any ...

**13**

votes

**5**answers

974 views

### K_2 of rings of algebraic integers

Let $R$ be the ring of integers in an algebraic number field. There are beautiful descriptions of $K_0(R)$ and $K_1(R)$. Namely, $\tilde{K}_0(R)$ is the class group of $R$ and $K_1(R)$ is the group ...