A survey on relative Lipschitz saturation of algebras and its relation with radicial algebras

Autores/as

  • Thiago da Silva Universidade Federal do Espírito Santo
  • Guilherme Schultz Netto Colégio Marista Colatina

DOI:

https://doi.org/10.14244/lajm.v3i1.31

Palabras clave:

Lipschitz saturation, Commutative Algebra

Resumen

In this work, we introduce Lipman’s work on relative Lipschitz saturation, along with its key categorical and algebraic properties, and demonstrate how Lipman proved that such a structure always gives rise to a radicial algebra.

Citas

A. Altman and S. Kleiman, A term of commutative algebra, Cambridge, USA, 2021.

N. Bourbaki, Elements of mathematics - commutative algebra, Hermann, 1972.

T. da Silva and M. Ribeiro, Universally injective and integral contractions on relative Lipschitz saturation of algebras, Journal of Algebra, Vol. 662, 902-922 (2024). DOI: https://doi.org/10.1016/j.jalgebra.2024.08.024

T. Gaffney, Bi-Lipschitz equivalence, integral closure and invariants, Real and complex singularities, London Math. Soc. Lecture Note Ser., 380, Cambridge Univ. Press, Cambridge (2010), 125–137. DOI: https://doi.org/10.1017/CBO9780511731983.010

T. Gaffney, The genericity of the infinitesimal Lipschitz condition for hypersurfaces, Journal of Singularities, (10), 108-123 (2014). DOI: https://doi.org/10.5427/jsing.2014.10g

A. Grothendieck and J. Dieudonné, El´éments de géométrie algébrique, Springer-Verlag, Berlim, 1971.

M. Lejeune and B. Teissier, Clˆôture int´égrale des id´éaux et ´équisingularit´é, Ann. Fac. Toulouse Math. (6), 17, no.4, 781-859 (2008). DOI: https://doi.org/10.5802/afst.1203

J. Lipman, Relative Lipschitz saturation, American Journal of Mathematics, Vol. 97 (3), 791-813 (1975). DOI: https://doi.org/10.2307/2373777

F. Pham and B. Teissier, Fractions Lipschitziennes d’une alg`ébre analytique complexe et saturation de Zariski (June 1969). 42 pages. Ce travail est la base de l’expos´é de Frédéric Pham au Congrès International des Mathématiciens, Nice 1970.

O. Zariski, Studies in equisingularity I: Equivalent singularities of plane algebroid curves, American Journal of Mathematics, Vol. 87, No. 2, pp. 507-536 (1965). DOI: https://doi.org/10.2307/2373019

O. Zariski, Studies in equisingularity II: Equisingularity in codimension 1 (and characteristic zero), American Journal of Mathematics, Vol. 87, No. 4, pp. 972-1006 (1965). DOI: https://doi.org/10.2307/2373257

O. Zariski, Studies in equisingularity III: Saturation of local rings and equisingularity, American Journal of Mathematics, vol. 90, No. 3, pp. 961-1023 (1968). DOI: https://doi.org/10.2307/2373492

O. Zariski, General theory of saturation and of saturated local rings I: Saturation of complete local domains of dimension one having arbitrary coefficient fields (of characteristic zero), American Journal of Mathematics, Vol. 93, No. 3, pp. 573-648 (1971). DOI: https://doi.org/10.2307/2373462

O. Zariski, General theory of saturation and of saturated local rings II: Saturared local rings of dimension 1, American Journal of Mathematics, Vol. 93, No. 4, pp. 872-964 (1971). DOI: https://doi.org/10.2307/2373741

O. Zariski, General theory of saturation and of saturated local rings III: Saturation in arbitrary dimension and, in particular, saturation of algebroid hypersurfaces, American Journal of Mathematics, Vol. 97, No. 2, pp. 415-502 (1975). DOI: https://doi.org/10.2307/2373720

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Publicado

2024-09-28

Cómo citar

[1]
da Silva, T. y Schultz Netto, G. 2024. A survey on relative Lipschitz saturation of algebras and its relation with radicial algebras. Latin American Journal of Mathematics. 3, 1 (sep. 2024), 1–18. DOI:https://doi.org/10.14244/lajm.v3i1.31.

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