Teoria dos mÃdulos idealizadores diferenciais

AUTOR(ES)

Cleto Brasileiro Miranda Neto

DATA DE PUBLICAÇÃO

2006

RESUMO

Given an ideal in a polynomial ring (with coefficients in a field usually assumed to have characteristic zero), we may consider the derivations that preserve it. They give rise to a special module called differential idealizer (of the given ideal). Such an object plays a primordial role in this thesis, which is divided into two main sections. In the first section the theory of such modules is developed from a completely general definition: we propose a relative version, not necessarily polynomial, with properties and techniques that turn out to be useful to several subsequent results. We then focus on polynomial idealizers, mainly giving effective criteria for reflexiveness and freeness, as well as introducing the class of the so-called differentially free ideals (and rings) (non-trivial generalization of the well-known notion of free divisor). The second section deals with applications to the classical module of derivations (or of tangent vector fields) of an algebra of finite type over a field. Firstly a computational method to obtain a set of generators is given. Obstructions to its Cohen-Macaulayness are investigated - one of them being that the ring must be equidimensional -, with criteria in the case of hypersurfaces and homogeneous complete intersections with isolated singularity. Primary decomposition in the reduced case, blowup algebras in the hypersurface case and certain multiplicity estimates are established. Finally, a free resolution in the case of differentially free rings is explicited, and versions of the Zariski-Lipman Conjecture are settled.

ASSUNTO(S)

cohen-macaulicidade differentially free ring, cohen-macaulayness hipersuperfÃcie blowup algebra, zariski-lipmanv hypersurface derivaÃÃo divisor livre anel diferencialmente livre Ãlgebra de explosÃo derivation, differential idealizer, free divisor matematica idealizador diferencial zariski-lipman.

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