Centers and Azumaya loci for finite W-algebras in positive characteristic

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Zariski compactness of minimal spectrum and flat compactness of maximal spectrum

In this paper, Zariski compactness of the minimal spectrum and flat compactness of the maximal spectrum are characterized.

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Let M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph $G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

Some Applications

This chapter describes some applications of the theory developed in the previous chapters in a variety of different mathematical contexts. The main methodology used to generate such applications is the ‘bridge technique’ presented in Chapter 2. The discussed topics include restrictions of Morita equivalences to quotients of the two theories involved, give a solution to a prozblem of Lawvere concerning the boundary operator on subtoposes, establish syntax-semantics ‘bridges’ for quotients of theories of presheaf type, present topos-theoretic interpretations and generalizations of Fraïssé’s theorem in model theory on countably categorical theories and of topological Galois theory, develop a notion of maximal spectrum of a commutative ring with unit and investigate compactness conditions for geometric theories allowing one to identify theories lying in smaller fragments of geometric logic.

A characterization of commutative rings whose maximal ideal spectrum is Noetherian

An ideal [Formula: see text] of a ring [Formula: see text] is called pseudo-irreducible if [Formula: see text] cannot be written as an intersection of two comaximal proper ideals of [Formula: see text]. In this paper, it is shown that the maximal spectrum of [Formula: see text] is Noetherian if and only if every proper ideal of [Formula: see text] can be expressed as a finite intersection of pseudo-irreducible ideals. Using a result of Hochster, we characterize all [Formula: see text] quasi-compact Noetherian topological spaces.

ON GORENSTEINNESS OF HOPF MODULE ALGEBRAS

Let H be a Hopf algebra with a bijective antipode, A an H-simple H-module algebra finitely generated as an algebra over the ground field and module-finite over its centre. The main result states that A has finite injective dimension and is, moreover, Artin–Schelter Gorenstein under the additional assumption that each H-orbit in the space of maximal ideals of A is dense with respect to the Zariski topology. Further conclusions are derived in the cases when the maximal spectrum of A is a single H-orbit or contains an open dense H-orbit.