Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Torsion freeness of symmetric powers of ideals

Author: Alexandre B. Tchernev
Journal: Trans. Amer. Math. Soc. 359 (2007), 3357-3367
MSC (2000): Primary 13C12, 13D30, 13A30
Published electronically: January 26, 2007
MathSciNet review: 2299459
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ I$ be an ideal in a Noetherian commutative ring $ R$ with unit, let $ k\ge 2$ be an integer, and let $ \alpha_k : S_k I\longrightarrow I^k$ be the canonical surjective $ R$-module homomorphism from the $ k$th symmetric power of $ I$ to the $ k$th power of $ I$. When $ \mathrm{pd}_R I\le 1$ or when $ I$ is a perfect Gorenstein ideal of grade $ 3$, we provide a necessary and sufficient condition for $ \alpha_k$ to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of $ I$. When $ I=\mathfrak{m}$ is a maximal ideal of $ R$ we show that $ \alpha_k$ is an isomorphism if and only if $ R_{\mathfrak{m}}$ is a regular local ring. In all three cases for $ I$ our results yield that if $ \alpha_k$ is an isomorphism, then $ \alpha_t$ is also an isomorphism for each $ 1\le t\le k$.

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

  • 1. L. Avramov, Complete intersections and symmetric algebras, J. Algebra 73 (1981), 248-263. MR 0641643 (83e:13024)
  • 2. G. Boffi and R. Sánchez, On the resolutions of the powers of the Pfaffian ideal, J. Algebra 152 (1992), 463-491. MR 1194315 (93j:14065)
  • 3. W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Math. 1327, Springer-Verlag, Berlin 1988. MR 0953963 (89i:13001)
  • 4. D. Buchsbaum and D. Eisenbud, Algebraic structures for finite free resolutions and some structure theorems for ideals of codimension $ 3$, Amer. J. Math. 99 (1977), 447-485. MR 0453723 (56:11983)
  • 5. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, New York 1995. MR 1322960 (97a:13001)
  • 6. A. Kustin and B. Ulrich, A family of complexes associated to an almost alternating map, with applications to residual intersections, Mem. Amer. Math. Soc. 95 (1992), no. 461. MR 1091668 (92i:13012)
  • 7. A. Micali and N. Roby, Algèbres symétriques et syzygies, J. Algebra 17 (1971), 460-469. MR 0282964 (44:198)
  • 8. A. Micali, P. Salmon, and P. Samuel, Integrité et factorialité des algèbres symétriques, Atas do IV Coloquio Brasileiro de Matematica, Sao Paolo 1965. MR 0207741 (34:7556)
  • 9. F. Planas-Vilanova, Rings of weak dimension one and syzygetic ideals, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3015-3017. MR 1328371 (96m:13005)
  • 10. T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconcelos, J. Algebra 99 (1986), 458-465. MR 0837555 (87i:13006)
  • 11. A. Simis and W. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), 203-224. MR 0610474 (82i:13016)
  • 12. A. Tchernev, Acyclicity criteria for complexes associated with an alternating map, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2861-2869. MR 1840088 (2002d:13017)
  • 13. W. Vasconcelos, Arithmetic of blowup algebras, Cambridge University Press, Cambridge 1994. MR 1275840 (95g:13005)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13C12, 13D30, 13A30

Retrieve articles in all journals with MSC (2000): 13C12, 13D30, 13A30

Additional Information

Alexandre B. Tchernev
Affiliation: Department of Mathematics, University at Albany, SUNY, Albany, New York 12222

Keywords: Torsion freeness, symmetric algebra, Rees algebra, symmetric powers
Received by editor(s): October 29, 2004
Received by editor(s) in revised form: July 5, 2005
Published electronically: January 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society