The structure of the inverse system of level K-algebras

• Autores: Shreedevi K. Masuti, Laura Tozzo
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 3, 2018, págs. 451-477
• Idioma: inglés
• DOI: 10.1007/s13348-018-0212-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • Macaulay’s inverse system is an effective method to construct Artinian K-algebras with the additional properties of being, for example, Gorenstein, level, or having any specific socle type. Recently, Elias and Rossi (Adv Math 314:306–327, 2017) gave the structure of the inverse system of d-dimensional Gorenstein K-algebras for any d>0. In this paper we extend their result by establishing a one-to-one correspondence between d-dimensional level K-algebras and suitable submodules of the divided power ring. We give several examples to illustrate our result.

• Referencias bibliográficas
  • Bertella, V.: Hilbert function of local Artinian level rings in codimension two. J. Algebra 321, 1429–1442 (2009)
  • Boij, M.: Artin level algebras. Ph.D. thesis, Royal Institute of Technology, Stockholm (1994)
  • Boij, M.: Betti numbers of compressed level algebras. J. Pure Appl. Algebra 134(2), 111–131 (1999)
  • Boij, M.: Artin level modules. J. Algebra 226(1), 361–374 (2000)
  • Boij, M.: Reducible family of height three level algebras. J. Algebra 321(1), 86–104 (2009)
  • Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised edn. Cambridge University Press, Cambridge (1998)
  • Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)
  • Ciliberto, C.: Algebra lineare. Bollati Boringhieri, Turin (1994)
  • Conca, A.: Divisor class group and canonical class of determinantal rings defined by ideals of minors of a symmetric matrix. Arch. Math. 63(3),...
  • Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-1-0: a computer algebra system for polynomial computations. http://www.singular.uni-kl.de...
  • De Stefani, A.: Artinian level algebras of low socle degree. Commun. Algebra 42, 729–754 (2014)
  • Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
  • Elias, J., Iarrobino, A.: The Hilbert function of a Cohen–Macaulay local algebra: extremal Gorenstein algebras. J. Algebra 110(2), 344–356...
  • Elias, J.: Inverse-syst.lib: singular library for computing Macaulay’s inverse systems. www.ub.edu/C3A/elias/inverse-syst-v.5.2.lib. arXiv:1501.01786...
  • Elias, J., Rossi, M.E.: The structure of the inverse system of Gorenstein K-algebras. Adv. Math. 314, 306–327 (2017)
  • Emsalem, J.: Géométrie des points épais. Bull. Soc. Math. France 106(4), 399–416 (1978)
  • Fouli, L.: A study on the core of ideals. Ph.D. thesis, Purdue University (2006)
  • Fröberg, R.: Connections between a local ring and its associated graded ring. J. Algebra 111(2), 300–305 (1987)
  • Gabriel, P.: Objets injectifs dans les catégories abéliennes. Séminaire P. Dubriel 1958-1959 17, 1–32 (1959)
  • Geramita, A.V., Harima, T., Migliore, J.C., Shin, Y.S.: The Hilbert function of a level algebra. Mem. Am. Math. Soc. 186(872), vi+139...
  • Goto, S.: On the Gorensteinness of determinantal loci. J. Math. Kyoto Univ. 19(2), 371–374 (1979)
  • Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ (2016)
  • Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge...
  • Huneke, C., Trung, N.V.: On the core of ideals. Compos. Math. 141(1), 1–18 (2005)
  • Iarrobino, A.: Compressed algebras: artin algebras having given socle degrees and maximal length. Trans. Am. Math. Soc. 285(1), 337–378 (1984)
  • Iarrobino, A.: Associated graded algebra of a Gorenstein Artin algebra. Mem. Am. Math. Soc. 107(514), viii+115 (1994)
  • Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1994). (Revised...
  • Mantero, P., Xie, Y.: Generalized stretched ideals and Sally’s conjecture. J. Pure Appl. Algebra 220(3), 1157–1177 (2016)
  • Molinelli, S., Tamone, G.: On the Hilbert function of certain rings of monomial curves. J. Pure Appl. Algebra 101(2), 191–206 (1995)
  • Polini, C., Ulrich, B.: A formula for the core of an ideal. Math. Ann. 331(3), 487–503 (2005)
  • Rossi, M.E., Valla, G.: Cohen–Macaulay local rings of embedding dimension e+d-3. Proc. Lond. Math. Soc. (3) 80(1), 107–126 (2000)
  • Rossi, M.E., Valla, G.: Hilbert Functions of Filtered Modules. Lecture Notes of the Unione Matematica Italiana, vol. 9. Springer-Verlag, Berlin...
  • Stanley, R.: Cohen–Macaulay complexes. In: Higher combinatorics (Proceedings of NATO Advanced Study Institute, Berlin, 1976). NATO Advanced...
  • Stanley, R.: Combinatorics and Commutative Algebra. Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston Inc, Boston (1996)
  • Xie, Y.: Formulas for the multiplicity of graded algebras. Trans. Am. Math. Soc. 364(8), 4085–4106 (2012)