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Compressed algebras: Artin algebras having given socle degrees and maximal length


Author: Anthony Iarrobino
Journal: Trans. Amer. Math. Soc. 285 (1984), 337-378
MSC: Primary 13E10; Secondary 13H10, 14B07, 35E99
MathSciNet review: 748843
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Abstract: J. Emsalem and the author showed in [18] that a general polynomial $ f$ of degree $ j$ in the ring $ \mathcal{R} = k[ {{y_1},\ldots,{y_r}} ]$ has $ \left( {\begin{array}{*{20}{c}} {j + r - 1} \\ {r - 1} \\ \end{array} } \right)$ linearly independent partial derivates of order $ i$, for $ i = 0,1,\ldots,t = [ {j/2} ]$. Here we generalize the proof to show that the various partial derivates of $ s$ polynomials of specified degrees are as independent as possible, given the room available.

Using this result, we construct and describe the varieties $ G(E)$ and $ Z(E)$ parametrizing the graded and nongraded compressed algebra quotients $ A = R/I$ of the power series ring $ R = k[[{x_1},\ldots,{x_r}]]$, having given socle type $ E$. These algebras are Artin algebras having maximal length $ \dim {_{k}}A$ possible, given the embedding degree $ r$ and given the socle-type sequence $ E = ({e_1},\ldots,{e_s})$, where $ {e_i}$ is the number of generators of the dual module $ \hat A$ of $ A$, having degree $ i$. The variety $ Z(E)$ is locally closed, irreducible, and is a bundle over $ G(E)$, fibred by affine spaces $ {{\mathbf{A}}^N}$ whose dimension is known.

We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable--have no deformation to $ (k + \cdots + k)$--for dimension reasons. For some choices of the sequence $ E,{\text{D}}$. Buchsbaum, $ {\text{D}}$. Eisenbud and the author have shown that the graded compressed algebras of socle-type $ E$ have almost linear minimal resolutions over $ R$, with ranks and degrees determined by $ E$. Other examples have given type $ e = {\dim_k}\;({\text{socle}}\;A)$ and are defined by an ideal $ I$ with certain given numbers of generators in $ R = k[[{x_1},\ldots\;,{x_r}]]$.

An analogous construction of thin algebras $ A = R/({f_1},\ldots,{f_s})$ of minimal length given the initial degrees of $ {f_1},\ldots,{f_s}$ is compared to the compressed algebras. When $ r = 2$, the thin algebras are characterized and parametrized, but in general when $ r > 3$, even their length is unknown. Although $ k = {\mathbf{C}}$ through most of the paper, the results extend to characteristic $ p$.


References [Enhancements On Off] (What's this?)

  • [1] Rüdiger Achilles and Luchezar L. Avramov, Relations between properties of a ring and of its associated graded ring, Seminar D. Eisenbud/B. Singh/W. Vogel, Vol. 2, Teubner-Texte Math., vol. 48, Teubner, Leipzig, 1982, pp. 5—29. MR 686455
  • [2] Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, 10.1016/0001-8708(82)90039-1
  • [3] Rosalba Barattero and Elsa Zatini, Relations between the type of a point on an algebraic variety and the type of its tangent cone, J. Algebra 66 (1980), no. 2, 386–399. MR 593601, 10.1016/0021-8693(80)90094-0
  • [4] D. Bayer, The division algorithm and the Hilbert scheme, Thesis, Harvard University, 1982.
  • [5] David Berman, Simplicity of a vector space of forms: finiteness of the number of complete Hilbert functions, J. Algebra 45 (1977), no. 1, 88–93. MR 0450276
  • [6] -, Hilbert functions of vector spaces of forms, Ph.D. Dissertation, University of Texas, Austin, 1978.
  • [7] David Berman, The number of generators of a colength 𝑁 ideal in a power series ring, J. Algebra 73 (1981), no. 1, 156–166. MR 641638, 10.1016/0021-8693(81)90352-5
  • [8] Joël Briançon, Description de 𝐻𝑖𝑙𝑏ⁿ𝐶{𝑥,𝑦}, Invent. Math. 41 (1977), no. 1, 45–89. MR 0457432
  • [9] 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 0453723
  • [10] -, Almost linear resolutions, Appendix to D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra (to appear).
  • [11] R. Buchweitz, Deformations de diagrammes, deploiements et singularités très rigides, liaison algébrique, Thèse, Université de Paris VII, 1981.
  • [12] Jacques Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), no. 4, 399–416 (French, with English summary). MR 518046
  • [13] Gerd Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978), no. 1, 61–70 (German). MR 0480478
  • [14] M. Granger, Géométrie des schémas de Hilbert ponctuels, Thèse, Université de Nice, 1981.
  • [15] Edward L. Green, Complete intersections and Gorenstein ideals, J. Algebra 52 (1978), no. 1, 264–273. MR 0480472
  • [16] J. H. Grace and A. Young, The algebra of invariants, Cambridge Univ. Press, New York, 1903; reprint, Chelsea, New York.
  • [17] Ernst Kunz, Beispiel: Die kanonische Idealklasse eines eindimensionalen Cohen-Macaulay-Rings, Der kanonische Modul eines Cohen-Macaulay-Rings (Sem. Lokale Kohomologietheorie von Grothendieck, Univ. Regensburg, Regensburg, 1970/1971), Springer, Berlin, 1971, pp. 17–24, 103. Lecture Notes in Math., Vol. 238 (German). MR 0476727
  • [18] A. Iarrobino and J. Emsalem, Some zero-dimensional generic singularities; finite algebras having small tangent space, Compositio Math. 36 (1978), no. 2, 145–188. MR 515043
  • [19] A. Iarrobino, Reducibility of the families of 0-dimensional schemes on a variety, Invent. Math. 15 (1972), 72–77. MR 0301010
  • [20] A. Iarrobino, The number of generic singularities, Rice Univ. Studies 59 (1973), no. 1, 49–51. Complex analysis, 1972 (Proc. Conf., Rice Univ., Houston, Tex., 1972), Vol. I: Geometry of singularities. MR 0345967
  • [21] Anthony A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (1977), no. 188, viii+112. MR 0485867
  • [22] Antonio Iarrobino, Deformations of zero-dimensional complete intersections after M. Granger, T. Gaffney, Proceedings of the Week of Algebraic Geometry (Bucharest, 1980) Teubner-Texte zur Math., vol. 40, Teubner, Leipzig, 1981, pp. 92–105. MR 712518
  • [23] Anthony Iarrobino, Deforming complete intersection Artin algebras. Appendix: Hilbert function of 𝐶[𝑥,𝑦]/𝐼, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 593–608. MR 713096
  • [24] -, Ancestor ideals of vector spaces of forms, preprint, 1975.
  • [25] Hans Kleppe, Deformation of schemes defined by vanishing of Pfaffians, J. Algebra 53 (1978), no. 1, 84–92. MR 0498556
  • [26] 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
  • [27] F. S. Macaulay, On a method of dealing with the intersections of plane curves, Trans. Amer. Math. Soc. 5 (1904), no. 4, 385–410. MR 1500679, 10.1090/S0002-9947-1904-1500679-1
  • [28] Guerino Mazzola, Generic finite schemes and Hochschild cocycles, Comment. Math. Helv. 55 (1980), no. 2, 267–293. MR 576606, 10.1007/BF02566686
  • [29] David Mumford, Pathologies IV, Amer. J. Math. 97 (1975), no. 3, 847–849. MR 0460338
  • [30] Judith D. Sally, Stretched Gorenstein rings, J. London Math. Soc. (2) 20 (1979), no. 1, 19–26. MR 545198, 10.1112/jlms/s2-20.1.19
  • [31] Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
  • [32] Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 0485835
  • [33] E. K. Wakeford, On canonical forms, Proc. London Math. Soc. (2) 18 (1918), 403-410.
  • [34] Peter Schenzel, Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe, J. Algebra 64 (1980), no. 1, 93–101 (German). MR 575785, 10.1016/0021-8693(80)90136-2
  • [35] J. Herzog and E. Kunz, On the deviation and the type of a Cohen-Macaulay ring, Manuscripta Math. 9 (1973), 383–388. MR 0330155
  • [36] David J. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986), no. 1, 235–259. MR 839581, 10.1016/0021-8693(86)90076-1
  • [37] R. Fröberg and D. Laksov, Extremal Cohen-Macaulay rings, and Gorenstein rings, preprint, Stockholm University (to appear).
  • [38] I. Elias and A. Iarrobino, Extremal Gorenstein algebras of codimension three. Appendix: Hilbert function of compressed algebras, preprint, 1984.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0748843-4
Keywords: Artin algebra, Gorenstein algebra, Hilbert function, socle, type, generators, dualizing module, nonsmoothable algebras, minimal resolutions, almost linear resolutions, derivatives of polynomials, linearly independent dérivates, maximal rank, zero-dimensional schemes, parametrization, Hilbert scheme, deformation, irreducible, Hankel matrix, cactalecticant, invariants, power sum decomposition, forms, general polynomials, unking, compressed algebra
Article copyright: © Copyright 1984 American Mathematical Society