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Absolutely S-domains and pseudo-polynomial rings

Noomen Jarboui, Ihsen Yengui (2002)

Colloquium Mathematicae

A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V, the extension R/(q ∩ R) ⊆ V/q is algebraic. A Noetherian domain R is an AS-domain if and only if dim (R) ≤ 1. In Section 2, we study a class of R-subalgebras of R[X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial...

Actions of Hopf algebras on pro-semisimple noetherian algebras and their invariants

Andrzej Tyc (2001)

Colloquium Mathematicae

Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants A H is a noetherian Cohen-Macaulay...

Actions of the additive group G a on certain noncommutative deformations of the plane

Ivan Kaygorodov, Samuel A. Lopes, Farukh Mashurov (2021)

Communications in Mathematics

We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1 , both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y ∣ y x - x y = h ( x ) 〉 , where h is an arbitrary polynomial in x . In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽 [ t ] comodule algebra structures on A h . We also compute the Makar-Limanov invariant of absolute constants...

Adic-completion and some dual homological results.

Anne-Marie Simon (1992)

Publicacions Matemàtiques

Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.

Affine rulings of weighted projective planes

Daniel Daigle (2001)

Annales Polonici Mathematici

It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)

AK-invariant, some conjectures, examples and counterexamples

L. Makar-Limanov (2001)

Annales Polonici Mathematici

In my talk I am going to remind you what is the AK-invariant and give examples of its usefulness. I shall also discuss basic conjectures about this invariant and some positive and negative results related to these conjectures.

Algebraic characterization of finite (branched) coverings

M. Mulero (1998)

Fundamenta Mathematicae

Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism...

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