Journal of Commutative Algebra

A structure theorem for most unions of complete intersections

Alfio Ragusa and Giuseppe Zappalà

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Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are a union of complete intersections, we give a structure theorem for the arithmetically Cohen-Macaulay union of two complete intersections of codimension~2, of type $(d_1,e_1)$ and $(d_2,e_2)$ such that $\min \{d_1,e_1\}\ne \min \{d_2,e_2\}$. We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.

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J. Commut. Algebra, Volume 9, Number 3 (2017), 423-439.

First available in Project Euclid: 1 August 2017

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Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Almost complete intersections Gorenstein rings pfaffians Betti numbers


Ragusa, Alfio; Zappalà, Giuseppe. A structure theorem for most unions of complete intersections. J. Commut. Algebra 9 (2017), no. 3, 423--439. doi:10.1216/JCA-2017-9-3-423.

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  • D.A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447–485.
  • A. Cayley, Sur les déterminants gauches, Crelle's J. 38 (1854), 93–96.
  • S. Diesel, Irreducibility and dimension theorems for families of height $3$ Gorenstein algebras, Pacific J. Math. 172 (1996), 365–397.
  • F. Gaeta, Quelques progrès récents dans la classification des variétés algébriques d'un espace projectif, Deuxième Colloque de Géométrie Algébrique, Liège, 1952.
  • P. Heymans, Pfaffians and skew-symmetric matrices, Proc. Lond. Math. Soc. 19 (1969), 730–768.
  • C. Huneke and J. Koh, Resolutions of almost complete intersections, J. Algebra 145 (1992), 22–-31.
  • A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lect. Notes Math. 1721, Springer-Verlag, New York, 1999.
  • J. Migliore, Introduction to liaison theory and deficency modules, Progr. Math. 165, Birkhauser, Boston, 1998.
  • J. Migliore and R.M. Miró-Roig, On the minimal free resolution of $n+1$ general forms, Trans. Amer. Math. Soc. 355 (2003), 1–36.
  • J. Migliore, R.M. Miró-Roig and U. Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), 229-–257.
  • C. Peskine and L. Szpiro, Liaison des variétés algébriques, I, Invent. Math. 26 (1974), 271–302.
  • A. Ragusa and G. Zappalà, Properties of $3$-codimensional Gorenstein schemes, Comm. Algebra 29 (2001), 303–318.
  • ––––, Complete intersections containing Cohen Macaulay and Gorenstein schemes Alg. Colloq. 18 (2011), 857–872.
  • A. Ragusa and G. Zappalà, On the structure of resolutions of almost complete intersections, Int. J. Pure Appl. Math. 83 (2013), 13–23.
  • ––––, Characterization of the graded Betti numbers for almost complete intersections, Comm. Alg. 41 (2013), 492–-506.
  • S. Seo, Almost complete intersections, J. Algebra 320 (2008), 2594–2609.