A constructive approach to one-dimensional Gorenstein 𝐤-algebras

Elias, J.; Rossi, M.

doi:10.1090/tran/8376

A constructive approach to one-dimensional Gorenstein ${\mathbf {k}}$-algebras
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by J. Elias and M. E. Rossi PDF

Trans. Amer. Math. Soc. 374 (2021), 4953-4971 Request permission

Abstract:

Let $R$ be the power series ring or the polynomial ring over a field ${\mathbf {k}}$ and let $I$ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein ${\mathbf {k}}$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma .$ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein ${\mathbf {k}}$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay’s correspondence characterizing the $R$-submodules of $\Gamma$ in one-to-one correspondence with Gorenstein d-dimensional ${\mathbf {k}}$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein ${\mathbf {k}}$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma$. Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.

References

Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
Young Hyun Cho and Anthony Iarrobino, Inverse systems of zero-dimensional schemes in $\Bbb {P}^n$, J. Algebra 366 (2012), 42–77. MR 2942643, DOI 10.1016/j.jalgebra.2012.04.032
Aldo Conca, Maria Evelina Rossi, and Giuseppe Valla, Gröbner flags and Gorenstein algebras, Compositio Math. 129 (2001), no. 1, 95–121. MR 1856025, DOI 10.1023/A:1013160203998
Gert-Martin Greuel, Description of SINGULAR: a computer algebra system for singularity theory, algebraic geometry and commutative algebra, Euromath Bull. 2 (1996), no. 1, 161–172. MR 1413182
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
D. Eisenbud, R. Hartshorne, and F.-O. Schreyer, Twenty points in $\Bbb P^3$, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 180–199. MR 3380449
Joan Elias, On the analytic equivalence of curves, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 57–64. MR 838653, DOI 10.1017/S0305004100065865
J. Elias, Inverse-syst.lib–Singular library for computing Macaulay’s inverse systems, arXiv:1501.01786, 2015. http://www.ub.edu/C3A/elias/inverse-syst-v.5.2.lib.
J. Elias and M. E. Rossi, The structure of the inverse system of Gorenstein $k$-algebras, Adv. Math. 314 (2017), 306–327. MR 3658719, DOI 10.1016/j.aim.2017.04.025
J. Elias, M. E. Rossi, and G. Valla, On the Hilbert function of one-dimensional local complete intersections, J. Algebra 399 (2014), 489–515. MR 3144600, DOI 10.1016/j.jalgebra.2013.10.006
J. Elias and M. E. Rossi, Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4589–4604. MR 2922602, DOI 10.1090/S0002-9947-2012-05430-4
Jacques Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399–416 (French, with English summary). MR 518046
P. Gabriel, Objects injectifs dans les catégories abéliennes, Séminaire P. Dubriel 1958/59, 1959, pp. 17–01, 32.
Anthony V. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995) Queen’s Papers in Pure and Appl. Math., vol. 102, Queen’s Univ., Kingston, ON, 1996, pp. 2–114. MR 1381732
Shiro Goto, William Heinzer, and Mee-Kyoung Kim, The leading ideal of a complete intersection of height two. II, J. Algebra 312 (2007), no. 2, 709–732. MR 2333181, DOI 10.1016/j.jalgebra.2007.02.025
Mark D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17–76. MR 1256101, DOI 10.1023/A:1022450120589
Robin Hartshorne, Experiments with Gorenstein liaison, Matematiche (Catania) 55 (2000), no. 2, 305–318 (2002). Dedicated to Silvio Greco on the occasion of his 60th birthday (Catania, 2001). MR 1984203
Craig Huneke, Hyman Bass and ubiquity: Gorenstein rings, Algebra, $K$-theory, groups, and education (New York, 1997) Contemp. Math., vol. 243, Amer. Math. Soc., Providence, RI, 1999, pp. 55–78. MR 1732040, DOI 10.1090/conm/243/03686
Anthony A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), no. 514, viii+115. MR 1184062, DOI 10.1090/memo/0514
Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271, DOI 10.1007/BFb0093426
Anthony Iarrobino and Hema Srinivasan, Artinian Gorenstein algebras of embedding dimension four: components of $\Bbb P\textrm {Gor}(H)$ for $H=(1,4,7,\dots ,1)$, J. Pure Appl. Algebra 201 (2005), no. 1-3, 62–96. MR 2158748, DOI 10.1016/j.jpaa.2004.12.015
Jan O. Kleppe, Juan C. Migliore, Rosa Miró-Roig, Uwe Nagel, and Chris Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732, viii+116. MR 1848976, DOI 10.1090/memo/0732
Andrew R. Kustin and Matthew Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983), no. 2, 303–322. MR 725084, DOI 10.1016/0021-8693(83)90096-0
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
Shreedevi K. Masuti and Laura Tozzo, The structure of the inverse system of level $K$-algebras, Collect. Math. 69 (2018), no. 3, 451–477. MR 3842217, DOI 10.1007/s13348-018-0212-3
Juan C. Migliore and Chris Peterson, A construction of codimension three arithmetically Gorenstein subschemes of projective space, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3803–3821. MR 1432204, DOI 10.1090/S0002-9947-97-01978-8
D. G. Northcott, Injective envelopes and inverse polynomials, J. London Math. Soc. (2) 8 (1974), 290–296. MR 360555, DOI 10.1112/jlms/s2-8.2.290
Kristian Ranestad and Frank-Olaf Schreyer, The variety of polar simplices, Doc. Math. 18 (2013), 469–505. MR 3084557
Miles Reid, Gorenstein in codimension 4: the general structure theory, Algebraic geometry in east Asia—Taipei 2011, Adv. Stud. Pure Math., vol. 65, Math. Soc. Japan, Tokyo, 2015, pp. 201–227. MR 3380790, DOI 10.2969/aspm/06510201
Maria Evelina Rossi and Giuseppe Valla, Hilbert functions of filtered modules, Lecture Notes of the Unione Matematica Italiana, vol. 9, Springer-Verlag, Berlin; UMI, Bologna, 2010. MR 2723038, DOI 10.1007/978-3-642-14240-6