## Torsion freeness of symmetric powers of ideals

HTML articles powered by AMS MathViewer

- by Alexandre B. Tchernev PDF
- Trans. Amer. Math. Soc.
**359**(2007), 3357-3367 Request permission

## 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

- Luchezar L. Avramov,
*Complete intersections and symmetric algebras*, J. Algebra**73**(1981), no. 1, 248–263. MR**641643**, DOI 10.1016/0021-8693(81)90357-4 - Giandomenico Boffi and Rafael Sánchez,
*On the resolutions of the powers of the Pfaffian ideal*, J. Algebra**152**(1992), no. 2, 463–491. MR**1194315**, DOI 10.1016/0021-8693(92)90044-M - Winfried Bruns and Udo Vetter,
*Determinantal rings*, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR**953963**, DOI 10.1007/BFb0080378 - 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 - 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 - Andrew R. Kustin and Bernd Ulrich,
*A family of complexes associated to an almost alternating map, with applications to residual intersections*, Mem. Amer. Math. Soc.**95**(1992), no. 461, iv+94. MR**1091668**, DOI 10.1090/memo/0461 - Artibano Micali and Norbert Roby,
*Algèbres symétriques et syzygies*, J. Algebra**17**(1971), 460–469 (French). MR**282964**, DOI 10.1016/0021-8693(71)90002-0 - A. Micali, P. Salmon, and P. Samuel,
*Intégrité et factorialité des algèbres symétriques*, Proc. Fourth Brazilian Math. Colloq. (1963) (Portuguese), Conselho Nacional de Pesquisas, São Paulo, 1965, pp. 61–76 (French). MR**0207741** - Francesc Planas-Vilanova,
*Rings of weak dimension one and syzygetic ideals*, Proc. Amer. Math. Soc.**124**(1996), no. 10, 3015–3017. MR**1328371**, DOI 10.1090/S0002-9939-96-03416-8 - Timothy Porter,
*Homology of commutative algebras and an invariant of Simis and Vasconcelos*, J. Algebra**99**(1986), no. 2, 458–465. MR**837555**, DOI 10.1016/0021-8693(86)90038-4 - A. Simis and W. V. Vasconcelos,
*The syzygies of the conormal module*, Amer. J. Math.**103**(1981), no. 2, 203–224. MR**610474**, DOI 10.2307/2374214 - Alexandre B. Tchernev,
*Acyclicity criteria for complexes associated with an alternating map*, Proc. Amer. Math. Soc.**129**(2001), no. 10, 2861–2869. MR**1840088**, DOI 10.1090/S0002-9939-01-05935-4 - Wolmer V. Vasconcelos,
*Arithmetic of blowup algebras*, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR**1275840**, DOI 10.1017/CBO9780511574726

## Additional Information

**Alexandre B. Tchernev**- Affiliation: Department of Mathematics, University at Albany, SUNY, Albany, New York 12222
- MR Author ID: 610821
- Email: tchernev@math.albany.edu
- Received by editor(s): October 29, 2004
- Received by editor(s) in revised form: July 5, 2005
- Published electronically: January 26, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 3357-3367 - MSC (2000): Primary 13C12, 13D30, 13A30
- DOI: https://doi.org/10.1090/S0002-9947-07-04135-9
- MathSciNet review: 2299459