Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
-
Yairon Cid-Ruiz
,
Claudia Polini
Abstract
We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein rings. As a consequence, we obtain a new general framework to investigate blowup algebras. We use our results to study and determine the defining equations of the Rees algebra for certain families of ideals.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2201110
Award Identifier / Grant number: DMS-2201149
Funding statement: Claudia Polini was partially supported by NSF grant DMS-2201110. Bernd Ulrich was partially supported by NSF grant DMS-2201149.
Acknowledgements
We thank the reviewer for carefully reading our paper and for several comments and corrections. Yairon Cid-Ruiz is grateful to the mathematics departments of Purdue University and the University of Notre Dame, where the majority of the work was done, for their hospitality and for excellent working conditions.
References
[1] R. Apéry, Sur certains caractères numériques d’un idéal sans composant impropre, C. R. Acad. Sci. Paris 220 (1945), 234–236. Suche in Google Scholar
[2] W. Bruns, The canonical module of a determinantal ring, Commutative algebra: Durham 1981 (Durham 1981), London Math. Soc. Lecture Note Ser. 72, Cambridge University, Cambridge (1982), 109–120. 10.1017/CBO9781107325517.010Suche in Google Scholar
[3] W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge University, Cambridge 1993. Suche in Google Scholar
[4] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. 1327, Springer, Berlin 1988. 10.1007/BFb0080378Suche in Google Scholar
[5] R.-O. Buchweitz, Contributions à la théorie des singularités, Thèse d’ Etat, Université Paris VII, 1981. Suche in Google Scholar
[6] L. Busé, Elimination theory in codimension one and applications., Notes of lectures given at the CIMPA-UNESCO-IRAN school in Zanjan, Iran 2005. Suche in Google Scholar
[7] L. Busé, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. 10.1016/j.jalgebra.2009.01.030Suche in Google Scholar
[8] L. Busé, F. Catanese and E. Postinghel, Algebraic curves and surfaces—a history of shapes, SISSA Springer Ser. 4, Springer, Cham 2023. 10.1007/978-3-031-24151-2Suche in Google Scholar
[9] L. Busé and M. Chardin, Fibers of rational maps and elimination matrices: An application oriented approach, Commutative algebra, Springer, Cham (2021), 189–217. 10.1007/978-3-030-89694-2_6Suche in Google Scholar
[10] L. Busé, Y. Cid-Ruiz and C. D’Andrea, Degree and birationality of multi-graded rational maps, Proc. Lond. Math. Soc. (3) 121 (2020), no. 4, 743–787. 10.1112/plms.12336Suche in Google Scholar
[11] Y. Cid-Ruiz, Mixed multiplicities and projective degrees of rational maps, J. Algebra 566 (2021), 136–162. 10.1016/j.jalgebra.2020.08.037Suche in Google Scholar
[12] T. Cortadellas Benítez and C. D’Andrea, Rational plane curves parameterizable by conics, J. Algebra 373 (2013), 453–480. 10.1016/j.jalgebra.2012.09.034Suche in Google Scholar
[13]
T. Cortadellas Benítez and C. D’Andrea,
Minimal generators of the defining ideal of the Rees algebra associated with a rational plane parametrization with
[14] D. Cox, J. W. Hoffman and H. Wang, Syzygies and the Rees algebra, J. Pure Appl. Algebra 212 (2008), no. 7, 1787–1796. 10.1016/j.jpaa.2007.11.006Suche in Google Scholar
[15] D. A. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1–3, 23–36. 10.1016/j.tcs.2007.10.012Suche in Google Scholar
[16] D. A. Cox, Reflections on elimination theory, ISSAC’20—Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, ACM, New York (2020), 1–4. 10.1145/3373207.3403977Suche in Google Scholar
[17] D. Eisenbud, Commutative algebra, Grad. Texts in Math. 150, Springer, New York 1995. 10.1007/978-1-4612-5350-1Suche in Google Scholar
[18] D. Eisenbud and B. Ulrich, Duality and socle generators for residual intersections, J. reine angew. Math. 756 (2019), 183–226. 10.1515/crelle-2017-0045Suche in Google Scholar
[19] F. Gaeta, Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif, Deuxième colloque de géométrie algébrique (Liège 1952), Georges Thone, Liège (1952), 145–183. Suche in Google Scholar
[20] M. Herrmann, S. Ikeda and U. Orbanz, Equimultiplicity and blowing up, Springer, Berlin 1988. 10.1007/978-3-642-61349-4Suche in Google Scholar
[21] J. Herzog and A. Rahimi, Local duality for bigraded modules, Illinois J. Math. 51 (2007), no. 1, 137–150. 10.1215/ijm/1258735329Suche in Google Scholar
[22] J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento 1981), Heidelberger Taschenb., Dekker, New York (1983), 79–169. Suche in Google Scholar
[23] J. Herzog, A. Simis and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661–683. 10.1090/S0002-9947-1984-0737891-6Suche in Google Scholar
[24] J. Herzog, A. Simis and W. V. Vasconcelos, On the canonical module of the Rees algebra and the associated graded ring of an ideal, J. Algebra 105 (1987), no. 2, 285–302. 10.1016/0021-8693(87)90194-3Suche in Google Scholar
[25] J. Herzog, W. V. Vasconcelos and R. Villarreal, Ideals with sliding depth, Nagoya Math. J. 99 (1985), 159–172. 10.1017/S0027763000021553Suche in Google Scholar
[26] M. Hochster and J. A. Eagon, Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. 10.2307/2373744Suche in Google Scholar
[27] J. Hong, A. Simis and W. V. Vasconcelos, The equations of almost complete intersections, Bull. Braz. Math. Soc. (N. S.) 43 (2012), no. 2, 171–199. 10.1007/s00574-012-0009-zSuche in Google Scholar
[28] C. Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043–1062. 10.2307/2374083Suche in Google Scholar
[29] C. Huneke, On the associated graded ring of an ideal, Illinois J. Math. 26 (1982), no. 1, 121–137. 10.1215/ijm/1256046904Suche in Google Scholar
[30] C. Huneke, Strongly Cohen–Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), no. 2, 739–763. 10.1090/S0002-9947-1983-0694386-5Suche in Google Scholar
[31] C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200–210. 10.1016/0021-8693(86)90023-2Suche in Google Scholar
[32] C. Huneke and B. Ulrich, Residual intersections, J. reine angew. Math. 390 (1988), 1–20. 10.1515/crll.1988.390.1Suche in Google Scholar
[33] M. R. Johnson, Second analytic deviation one ideals and their Rees algebras, J. Pure Appl. Algebra 119 (1997), no. 2, 171–183. 10.1016/S0022-4049(96)00021-7Suche in Google Scholar
[34] J.-P. Jouanolou, Singularités rationnelles du résultant, Algebraic geometry (Copenhagen 1978), Lecture Notes in Math. 732, Springer, Berlin (1979), 183–213. 10.1007/BFb0066645Suche in Google Scholar
[35] J.-P. Jouanolou, Idéaux résultants, Adv. Math. 37 (1980), no. 3, 212–238. 10.1016/0001-8708(80)90034-1Suche in Google Scholar
[36] J.-P. Jouanolou, Le formalisme du résultant, Adv. Math. 90 (1991), no. 2, 117–263. 10.1016/0001-8708(91)90031-2Suche in Google Scholar
[37] J.-P. Jouanolou, Aspects invariants de l’élimination, Adv. Math. 114 (1995), no. 1, 1–174. 10.1006/aima.1995.1042Suche in Google Scholar
[38] J.-P. Jouanolou, Résultant anisotrope, compléments et applications, Electron. J. Combin. 3 (1996), no. 2, Research Paper R2. 10.37236/1260Suche in Google Scholar
[39] J.-P. Jouanolou, Formes d’inertie et résultant: un formulaire, Adv. Math. 126 (1997), no. 2, 119–250. 10.1006/aima.1996.1609Suche in Google Scholar
[40] Y. Kim and V. Mukundan, Equations defining certain graphs, Michigan Math. J. 69 (2020), no. 4, 675–710. 10.1307/mmj/1580439628Suche in Google Scholar
[41] E. Kunz, Kähler differentials, Adv. Lect. Math., Friedrich Vieweg & Sohn, Braunschweig 1986. 10.1007/978-3-663-14074-0Suche in Google Scholar
[42] A. Kustin, C. Polini and B. Ulrich, The bi-graded structure of symmetric algebras with applications to Rees rings, J. Algebra 469 (2017), 188–250. 10.1016/j.jalgebra.2016.08.014Suche in Google Scholar
[43] A. R. Kustin, C. Polini and B. Ulrich, Rational normal scrolls and the defining equations of Rees algebras, J. reine angew. Math. 650 (2011), 23–65. 10.1515/crelle.2011.002Suche in Google Scholar
[44] A. R. Kustin, C. Polini and B. Ulrich, The equations defining blowup algebras of height three Gorenstein ideals, Algebra Number Theory 11 (2017), no. 7, 1489–1525. 10.2140/ant.2017.11.1489Suche in Google Scholar
[45] A. R. Kustin, C. Polini and B. Ulrich, Degree bounds for local cohomology, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1251–1267. 10.1112/plms.12364Suche in Google Scholar
[46] A. R. Kustin and B. Ulrich, If the socle fits, J. Algebra 147 (1992), no. 1, 63–80. 10.1016/0021-8693(92)90252-HSuche in Google Scholar
[47] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Grad. Texts in Math. 227, Springer, New York 2005. Suche in Google Scholar
[48] S. Morey, Equations of blowups of ideals of codimension two and three, J. Pure Appl. Algebra 109 (1996), no. 2, 197–211. 10.1016/0022-4049(95)00087-9Suche in Google Scholar
[49] S. Morey and B. Ulrich, Rees algebras of ideals with low codimension, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653–3661. 10.1090/S0002-9939-96-03470-3Suche in Google Scholar
[50] C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302. 10.1007/BF01425554Suche in Google Scholar
[51] C. Polini and B. Ulrich, Necessary and sufficient conditions for the Cohen–Macaulayness of blowup algebras, Compos. Math. 119 (1999), no. 2, 185–207. 10.1023/A:1001704003619Suche in Google Scholar
[52] G. Scheja and U. Storch, Über Spurfunktionen bei vollständigen Durchschnitten, J. reine angew. Math. 278(279) (1975), 174–190. 10.1515/crll.1975.278-279.174Suche in Google Scholar
[53] A. Simis, B. Ulrich and W. V. Vasconcelos, Cohen–Macaulay Rees algebras and degrees of polynomial relations, Math. Ann. 301 (1995), no. 3, 421–444. 10.1007/BF01446637Suche in Google Scholar
[54] B. Ulrich, Ideals having the expected reduction number, Amer. J. Math. 118 (1996), no. 1, 17–38. 10.1353/ajm.1996.0006Suche in Google Scholar
[55] B. Ulrich and W. V. Vasconcelos, The equations of Rees algebras of ideals with linear presentation, Math. Z. 214 (1993), no. 1, 79–92. 10.1007/BF02572392Suche in Google Scholar
[56] W. V. Vasconcelos, On the equations of Rees algebras, J. reine angew. Math. 418 (1991), 189–218. 10.1515/crll.1991.418.189Suche in Google Scholar
[57] W. V. Vasconcelos, Arithmetic of blowup algebras, London Math. Soc. Lecture Note Ser. 195, Cambridge University, Cambridge 1994. 10.1017/CBO9780511574726Suche in Google Scholar
[58] J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232. 10.1017/S002776300001566XSuche in Google Scholar
[59] H. Wiebe, Über homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274. 10.1007/BF01350771Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
