We prove the extended Hilbert’s Nullstellensatz in the context of Hu-Liu polynomial trirings. Which kinds of noncommutative rings are suitable for extending algebra geometry? Different attempts have been made to answer this question, but a satisfactory answer is still in hiding. My attempt at answering this question comes from the trivial extension of a ring by a bimodule over the ring. The tri...

Given a finite dimensional algebra A of finite global dimension, we consider the trivial extension of A by the A − A-bimodule ⊕i≥2 Ext 2 A(DA,A), which we call the higher relation bimodule. We first give a recipe allowing to construct the quiver of this trivial extension in case A is a string algebra and then apply it to prove that, if A is gentle, then the tensor algebra of the higher relation...

In this paper we study geometric Poisson brackets and we show that, if M = (G n IRn)/G endowed with an affine geometry (in the Klein sense), and if G is a classical Lie group, then the geometric Poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves, except for the case G = GL(n, IR). This trivial extension does not exist in other nonaffine cases (pr...

We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension. 0. Asymptotic dimension asdim of a metric space was defined by Gromov for studying asymptotic invariants of discrete groups [1]. This dimension can be considered as ...

This question is, I feel, in the same kind of idiot category as how long is a piece of string? interviewed recently by a journalist planning to write an article about the women (like me) who conduct 'Auntie' columns in women's magazines, I was asked what proportion of the letters I received each week could be labelled trivial. Her eyebrows shot up when I said that in my view, no human problem w...

Introduction Let ̃ SL(2,F) be the metaplectic two-fold cover of SL(2,F), the special linear group in two variables over a local field F of characteristic 0. The inverse image T̃ of a maximal torus T in ̃ SL(2,F) is an abelian extension of T by ±1. We consider the question of whether this extension is trivial. We exclude the case F = C, which is trivial. More generally suppose A is a subgroup of th...

A word is called unbordered, if it has no proper prefix which is also a suffix of that word. Let μ(w) denote the length of the longest unbordered factor of a word w. Let a word where the longest unbordered prefix is equal to μ(w) be called Duval extension. A Duval extension is called trivial, if its longest unbordered factor is of the length of the period of that Duval extension. In 1982 it was...

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We show how to produce SOn-equivariant vector fields on non-trivial SOn-torsors which correspond to quadratic forms non-equivalent to the unit form. For n 3 we then give an example of a Picard–Vessiot extension with group SOn which is the function field of a non-trivial SOn-torsor. © 2007 Elsevier Inc. All rights reserved.