F. S. Macaulay: From plane curves to Gorenstein rings

Eisenbud, David; Gray, Jeremy

doi:10.1090/bull/1787

F. S. Macaulay: From plane curves to Gorenstein rings
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by David Eisenbud and Jeremy Gray

Bull. Amer. Math. Soc. 60 (2023), 371-406

DOI: https://doi.org/10.1090/bull/1787

Published electronically: April 27, 2023

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Abstract:

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation.

Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

References

Macaulay, F.S., 1895, On point-groups in relation to curves, Proceedings of the London Mathematical Society (1) 26, 495–544.
Macaulay, F.S., 1895n, Geometrical conics, Cambridge, 2nd edition, 1906.
Macaulay, F.S., 1897n, On the deformation of a plane closed polygon so that a certain function remains constant, Proceedings of the London Mathematical Society (1) 28, 442–446.
Macaulay, F.S., 1898a, Point-groups in a plane, and their effect in determining algebraic curves, Proceedings of the London Mathematical Society (1) 29, 673–695.
Macaulay, F.S., 1898b, On the intersections of plane curves, Bulletin of the American Mathematical Society 4, 540–544.
Macaulay, F.S., 1899a, The theorem of residuation, Noether’s theorem, and the Riemann–Roch theorem, Proceedings of the London Mathematical Society (1) 31, 15–30.
Macaulay, F.S., 1899b, The theorem of residuation, being a general treatment of the intersections of plane curves at multiple points, Proceedings of the London Mathematical Society (1) 31, 381–423.
Macaulay, F.S., 1900, Extensions of the Riemann–Roch theorem in plane geometry, Proceedings of the London Mathematical Society (1) 32, 418–430.
Macaulay, F.S., 1900n, John Bolyai’s Science absolute of space, The Mathematical Gazette 1, 25–31, 49–60.
Macaulay, F.S., 1900nn, On continued fractions, The Mathematical Gazette 1, 39–40.
Macaulay, F.S., 1902, Some formulae in elimination, Proceedings of the London Mathematical Society (1) 35, 3–27.
Macaulay, F.S., 1904, On a method of dealing with the intersections of plane curves, Transactions of the American Mathematical Society 5, 385–410.
Macaulay, F.S., 1905, The intersections of plane curves, with extensions to $n$-dimension algebraic manifolds, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg, 284–312.
Macaulay, F.S., 1906n, Projective geometry, The Mathematical Gazette 3, 1–6.
Macaulay, F.S., 1906nn, On the axioms and postulates employed in the elementary plane constructions, The Mathematical Gazette 3, 78–81.
Macaulay, F.S., 1906nnn, On a problem in mechanics and the number of its solutions, The Mathematical Gazette 3, 365–373.
Macaulay, F.S., 1913, On the resolution of a given modular system into primary systems, including some properties of Hilbert numbers, Mathematische Annalen 74, 66–121.
Macaulay, F.S., 1915, On the algebraic theory of modular systems $($or modules of polynomials$)$, British Association Report 84, (Australia), 310–311.
Macaulay, F.S., 1916, The algebraic theory of modular systems, Cambridge Tracts in Mathematics, No. 19.
Macaulay, F.S., 1920–1923, Max Noether, Proceedings of the London Mathematical Society (2) 21, 37–42.
Macaulay, F.S., 1923, Note on the resultant of a number of polynomials of the same degree, Proceedings of the London Mathematical Society (2) 21, 14–21.
Macaulay, F.S., 1927, Some properties of enumeration in the theory of modular systems, Proceedings of the London Mathematical Society (2) 26, 531–555.
Macaulay, F.S., 1930n, Some inequalities connected with a method of representing positive integers, The Mathematical Gazette 15, 95–98.
Macaulay, F.S., 1930nn, W.J. Greenstreet, The Mathematical Gazette 15, 181–186.
Macaulay, F.S., 1932, Dr. Charlotte Angas Scott, Journal of the London Mathematical Society 7, 230–240.
Macaulay, F.S., 1934, Modern algebra and polynomial ideals, Proceedings of the Cambridge Philosophical Society 30, 27–46.
Shreeram Abhyankar, On Macaulay’s examples, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 1–16. Notes by A. Sathaye. MR 0466156
I. Bacharach, Ueber den Cayley’schen Schnittpunktsatz, Math. Ann. 26 (1886), no. 2, 275–299 (German). MR 1510347, DOI 10.1007/BF01444338
H. F. Baker, Francis Sowerby Macaulay, J. London Math. Soc. 13 (1938), no. 2, 157–160. MR 1574145, DOI 10.1112/jlms/s1-13.2.157
Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
E. Bertini, Zum Fundamentalsatz aus der Theorie der algebraischen Functionen, Math. Ann. 34 (1889), no. 3, 447–449 (German). MR 1510584, DOI 10.1007/BF01444051
Etienne Bézout, Théorie générale des équations algébriques, Ph.D. Pierres, 1779.
Umberto Bottazzini, The higher calculus: a history of real and complex analysis from Euler to Weierstrass, Springer-Verlag, New York, 1986. Translated from the Italian by Warren Van Egmond. MR 860945, DOI 10.1007/978-1-4612-4944-3
A. Brill and M. Noether, Die entwicklung der theorie der algebraischen funktionen in älterer und neuerer zeit, Deutsche Mathematiker-Vereinigung, 1894.
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
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
Giulio Caviglia and Alessio Sammartano, On the lex-plus-powers conjecture, Adv. Math. 340 (2018), 284–299. MR 3886170, DOI 10.1016/j.aim.2018.10.005
Arthur Cayley, The collected mathematical papers. Volume 1, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009. Reprint of the 1889 original. MR 2866537, DOI 10.1017/CBO9780511703775.090
Michel Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie, Éditions Jacques Gabay, Sceaux, 1989 (French). Reprint of the 1837 original. MR 1355544
Yairon Cid-Ruiz, Roser Homs, and Bernd Sturmfels, Primary ideals and their differential equations, Found. Comput. Math. 21 (2021), no. 5, 1363–1399. MR 4323620, DOI 10.1007/s10208-020-09485-6
Irvin Sol Cohen, THE STRUCTURE AND IDEAL THEORY OF LOCAL RINGS, ProQuest LLC, Ann Arbor, MI, 1942. Thesis (Ph.D.)–The Johns Hopkins University. MR 2937532
I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 16094, DOI 10.1090/S0002-9947-1946-0016094-3
Aldo Conca, Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math. 138 (1998), no. 2, 263–292. MR 1645574, DOI 10.1006/aima.1998.1740
R. C. Cowsik and M. V. Nori, On the fibres of blowing up, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 217–222 (1977). MR 572990
David Eisenbud and J. J. Gray, F. S. Macaulay: From geometry to modern commutative algebra. (to appear).
David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324. MR 1376653, DOI 10.1090/S0273-0979-96-00666-0
David Eisenbud and Frank-Olaf Schreyer, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579. With an appendix by Jerzy Weyman. MR 1969204, DOI 10.1090/S0894-0347-03-00423-5
Federico Gaeta, Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif, Deuxième Colloque de Géométrie Algébrique, Liège, 1952, Georges Thone, Liège; Masson & Cie, Paris, 1952, pp. 145–183 (French). MR 0052828
Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, 1993–present. Available at http://www.math.uiuc.edu/Macaulay2/.
Mark Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 76–86. MR 1023391, DOI 10.1007/BFb0085925
Curtis Greene and Daniel J. Kleitman, Proof techniques in the theory of finite sets, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 22–79. MR 513002
George-Henri Halphen, Mémoire sur la classification des courbes gauches algébriques, J. Éc. Polyt. 52 (1882), 1–200.
J. Hannak, Emmanuel Lasker, the life of a chess master, Dover, 1991. Translation from German of “Emmanuel Lasker, Biographie eines Schachweltmeisters”, Siegfried Engelhardt Verlag, Berlin-Frohnau, 1952.
David Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann. 36 (1890), no. 4, 473–534 (German). MR 1510634, DOI 10.1007/BF01208503
Craig Huneke, Powers of ideals generated by weak $d$-sequences, J. Algebra 68 (1981), no. 2, 471–509. MR 608547, DOI 10.1016/0021-8693(81)90276-3
Nazeran Idrees, Gerhard Pfister, and Afshan Sadiq, An algorithm to compute a primary decomposition of modules in polynomial rings over the integers, Studia Sci. Math. Hungar. 52 (2015), no. 1, 40–51. MR 3325942, DOI 10.1556/SScMath.52.2015.1.1296
Jean-Pierre Jouanolou, Aspects invariants de l’élimination, Adv. Math. 114 (1995), no. 1, 1–174 (French). MR 1344713, DOI 10.1006/aima.1995.1042
Irving Kaplansky, Commutative rings, Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
P. C. Kenschaft, Charlotte Angas Scott (1858–1931), in A century of mathematics in America. Part III, History of Mathematics, vol. 3, (P. Duren, H. M. Edwards, Uta C. Merzbach, eds.), American Mathematical Society, Providence, RI, pp. 241–252, 1989.
L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraische Grössen, J. Reine Angew. Math. 92 (1882), 1–122 (German). MR 1579896, DOI 10.1515/crll.1882.92.1
Wolfgang Krull, Dimensionstheorie in Stellenringen, J. Reine Angew. Math. 179 (1938), 204–226 (German). MR 1581594, DOI 10.1515/crll.1938.179.204
E. Lasker, Zur Theorie der moduln und Ideale, Math. Ann. 60 (1905), no. 1, 20–116 (German). MR 1511288, DOI 10.1007/BF01447495
F. S. MacAulay, Point-Groups in relation to Curves, Proc. Lond. Math. Soc. 26 (1894/95), 495–544. MR 1575910, DOI 10.1112/plms/s1-26.1.495
F. S. MacAulay, The Theorem of Residuation, being a general treatment of the Intersections of the Plane Curves at Multiple Points, Proc. Lond. Math. Soc. 31 (1899), 381–423. MR 1576720, DOI 10.1112/plms/s1-31.1.381
F.S. Macaulay, Extensions of the Riemann–Roch theorem in plane geometry, Proceedings of the London Mathematical Society 32 (1900), no. 1, 418–430.
F. S. MacAulay, Some Formulae in Elimination, Proc. Lond. Math. Soc. 35 (1903), 3–27. MR 1577000, DOI 10.1112/plms/s1-35.1.3
F. S. Macaulay, On the resolution of a given modular system into primary systems including some properties of Hilbert numbers, Math. Ann. 74 (1913), no. 1, 66–121. MR 1511752, DOI 10.1007/BF01455345
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
F. S. MacAulay, Note on the Resultant of a Number of Polynomials of the Same Degree, Proc. London Math. Soc. (2) 21 (1923), 14–21. MR 1575348, DOI 10.1112/plms/s2-21.1.14
F. S. MacAulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. London Math. Soc. (2) 26 (1927), 531–555. MR 1576950, DOI 10.1112/plms/s2-26.1.531
F. S. Macaulay, Some inequalities connected with a method of representing positive integers., Math. Gaz. 15 (1930), 95–98 (English).
F. S. Macaulay, Modern algebra and polynomial ideals, Proc. Cambridge Philos. Soc. 30 (1934), no. 1, 27–46. MR 4212331
T. T. Moh, On the unboundedness of generators of prime ideals in power series rings of three variables, J. Math. Soc. Japan 26 (1974), 722–734. MR 354680, DOI 10.2969/jmsj/02640722
A.F. Monna, Dirichlet’s principle: A mathematical comedy of errors and its influence on the development of analysis, Oosthoek, Scheltema & Holkema, 1975.
M. Noether, Ueber einen Satz aus der Theorie der algebraischen Functionen, Math. Ann. 6 (1873), 351–359.
M. Noether, Ueber den Fundamentalsatz der Theorie der algebraischen Functionen, Math. Ann. 30 (1887), no. 3, 410–417 (German). MR 1510452, DOI 10.1007/BF01443953
Alexander Ostrowski, Über vollständige Gebiete gleichmäßiger Konvergenz von Folgen analytischer Funktionen, Abh. Math. Sem. Univ. Hamburg 1 (1922), no. 1, 326–349 (German). MR 3069407, DOI 10.1007/BF02940599
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, DOI 10.1007/BF01425554
A. Prabhakar Rao, Liaison among curves in $\textbf {P}^{3}$, Invent. Math. 50 (1978/79), no. 3, 205–217. MR 520926, DOI 10.1007/BF01410078
J. Rosenthal, Emanuel Lasker: Vol. 1. Struggle and victories. World chess champion for 27 years, Exzelsior Verlag, Berlin, 2018. Translated from Der Mathematiker Emanuel Lasker, in Emanuel Lasker: Denker, Weltenbürger, Schachweltmeister, R. Forster, S. Hansen, M. Negele (eds.).
George Salmon, Higher algebra, Cambridge Library Collection, Metcalf and Son, Cambridge, 1885.
Richard P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142. MR 458437, DOI 10.1002/sapm1975542135
Ngô Viêt Trung, On the symbolic powers of determinantal ideals, J. Algebra 58 (1979), no. 2, 361–369. MR 540645, DOI 10.1016/0021-8693(79)90167-4
Bernd Ulrich, Artin-Nagata properties and reductions of ideals, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 373–400. MR 1266194, DOI 10.1090/conm/159/01519
B. L. van der Waerden, Moderne Algebra. Bd. I. Unter Benutzung von Vorlesungen von E. Artin und E. Noether., Vol. 23, Springer, Berlin, 1930 (German).
B. L. van der Waerden, Algebra. Teil II, Heidelberger Taschenbücher, Band 23, Springer-Verlag, Berlin-New York, 1967 (German). Unter Benutzung von Vorlesungen von E. Artin und E. Noether; Fünfte Auflage. MR 0233647
B. L. van der Waerden, On the sources of my book Moderne Algebra, Historia Math. 2 (1975), 31–40 (English, with German summary). MR 465721, DOI 10.1016/0315-0860(75)90034-8
A. Voss, Ueber einen fundamentalsatz aus der theorie der algebraischen functionen, Math. Ann. 27 (1887), 527–536.