We present an example of a subalgebra with infinite stable rank in the algebra of all bounded analytic functions in the unit disk.

In the classical Witt theory over a field F, the study of quadratic forms begins with two simple invariants: the dimension of a form modulo 2, called the dimension index and denoted e⁰: W(F) → ℤ/2, and the discriminant e¹ with values in k₁(F) = F*/F*², which behaves well on the fundamental ideal I(F)= ker(e⁰). Here a more sophisticated situation is considered, of quadratic forms over a scheme and, more generally, over an exact category with duality. Our purposes are: ...

Soit $E$ un enlacement de $n$ intervalles dans $D^2\times I$ d’extérieur $X$ et soit $X_0=X\cap D^2\times 0$. On utilise la propriété de la paire $(X,X_0)$ d’être $\Lambda$-acyclique pour certaines représentation $\rho$ de l’anneau du groupe fondamental $\pi$ de $X$ dans un anneau $\Lambda$ pour construire des invariants de torsion à valeurs dans le groupe $K_1\left(\Lambda \right)/\rho (\pm \pi )$. Un cas particulier est le polynôme d’Alexander en $n$ variables quand $\Lambda$ est l’anneau des fractions rationnelles $P/Q$ avec $Q(1,1,\cdots ,1)=1$ et $\rho$ est simplement l’abélianisation.

We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E\left(R\right)$ and a $u\in U\left(R\right)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann\left(a^{+}\right)$ and $u\in U\left(R\right)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR⟹aR\cong bR$.

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $L\left(1,E/\mathbb{Q}({\mu }_{p^n},\sqrt[p^n]{\Delta })\right).$ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\left({\mathbb{Z}}_p\left[\left[\mathrm{Gal}(\mathbb{Q}({\mu }_{{\mathrm{p}}^{\infty }},\sqrt[{\mathrm{p}}^{\infty }]{\Delta })/\mathbb{Q})\right]\right]\right)$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.

An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\left(\begin{array}{cc}a& b\\ & *\end{array}\right)\in M_2\left(S\right)$. The dual assertions are also proved.