Related papers: From Euler class groups to Mennicke symbols and a …

Let $R$ be a smooth affine domain of dimension $d\geq 2$ over an infinite perfect field $k$. We establish a morphism from the Euler class group $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$, the group of elementary orbits of unimodular rows.

Let $R$ be a reduced real affine algebra of (Krull) dimension $d \ge 2$ such that either $R$ has no real maximal ideals, or the intersection of all real maximal ideals in $R$ has height at least one. In this article, we prove the following:…

We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the…

We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type…

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,\Delta)$. These groups contain as special cases the odd dimensional general linear groups $GL_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic…

In this paper, we prove that there is group homomorphism from general linear group over a polynomial extension of a local ring to the W. van der Kallen group.

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…

Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic…

For each finite group G, we define the Grothendieck-Teichm\"uller group of G, denoted GT(G), and explore its properties. The theory of dessins d'enfants shows that the inverse limit of GT(G) as G varies can be identified with a group…

Let $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n \leq d\leq 2n-3$. Let $(a_1,...,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,...,a_n)$ has height $n$. Jean Fasel has associated to this row an…

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role…

Let A be a regular domain of dimension d containing an infinite field and let n be an integer with 2n\geq d+3. For a stably free A-module P of rank n, we prove that (i) P has a unimodular element if and only if the euler class of P is zero…

Mennicke--Newman lemma for unimodular rows was used by W. van der Kallen to give a group structure on the orbit set $\frac{Um_{n}(R)}{E_{n}(R)}$ for a commutative noetherian ring of dimension $d\leq 2n-4.$ In this paper, we generalise the…

We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We…

We introduce two families of diagrammatic monoidal supercategories. The first family, depending on an associative superalgebra, generalizes the oriented Brauer category. The second, depending on an involutive superalgebra, generalizes the…

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero…

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic…

We classify all finite 2-groups that have a cyclic or dihedral maximal subgroup and determine their automorphism groups. Based on this result, we classify all pairs $ (G,\mathcal{M}) $, such that $ G $ is a finite 2-group and $ \mathcal{M}…

We study the Euler characteristic of the real Milnor fibres of a real analytic map, using a relation between complex monodromy and complex conjugation. We deduce the result of Coste and Kurdyka that the Euler characteristic of the link of…