We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity.
In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices....
Soient un corps commutatif et un idéal de l’anneau des polynômes (éventuellement ). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme , élément de la clôture intégrale de l’idéal , on a une représentationoù .
We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...
We consider Stanley-Reisner rings where is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.
The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.