Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-03T01:42:50.319Z Has data issue: false hasContentIssue false

Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

Published online by Cambridge University Press:  09 November 2011

Hailong Dao*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA (email: hdao@math.ku.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Aus61]Auslander, M., Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631647.Google Scholar
[AB89]Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen–Macaulay approximations, Mém. Soc. Math. Fr. (N.S.) 38 (1989), 537.Google Scholar
[Bad78]Badescu, L., A remark on the Grothendieck–Lefschetz theorem about the Picard group, Nagoya Math. J. 71 (1978), 169179.CrossRefGoogle Scholar
[Cha99]Chan, C.-Y. J., Filtrations of modules, the Chow group, and the Grothendieck group, J. Algebra 219 (1999), 330344.Google Scholar
[Dao]Dao, H., Decency and rigidity over hypersurfaces, Trans. Amer. Math. Soc., to appear.Google Scholar
[Dao08]Dao, H., Some observations on local and projective hypersurfaces, Math. Res. Lett. 15 (2008), 207219.CrossRefGoogle Scholar
[Dao10]Dao, H., Remarks on non-commutative crepant resolutions of complete intersections, Adv. Math. 224 (2010), 10211030.CrossRefGoogle Scholar
[DLM10]Dao, H., Li, J. and Miller, C., On (non)rigidity of the Frobenius over Gorenstein rings, Algebra Number Theory 4–8 (2010), 10391053.CrossRefGoogle Scholar
[Del73]Deligne, P., Cohomologie étale : Séminaire de géométrie algébrique du Bois-Marie (SGA 7 II), Lecture Notes in Mathematics, vol. 340 (Springer, New York, 1973).Google Scholar
[Eis80]Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 3564.CrossRefGoogle Scholar
[Ful98]Fulton, W., Intersection theory (Springer, Berlin, 1998).Google Scholar
[Gab04]Gabber, O., On purity for the Brauer group, in Arithmetic algebraic geometry, Oberwolfach Report No. 37, Mathematisches Forschungsinstitut Oberwolfach (2004), 19751977.Google Scholar
[Har98]Hartshorne, R., Coherent functors, Adv. Math. 140 (1998), 4494.Google Scholar
[Hoc81]Hochster, M., The dimension of an intersection in an ambient hypersurface, in Proceedings of the First Midwest Algebraic Geometry Seminar (Chicago Circle, 1980), Lecture Notes in Mathematics, vol. 862 (Springer, New York, 1981), 93106.Google Scholar
[Hor64]Horrocks, G., Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc. (3) 14 (1964), 689713.Google Scholar
[HW97]Huneke, C. and Wiegand, R., Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), 161183.CrossRefGoogle Scholar
[HWJ01]Huneke, C., Wiegand, R. and Jorgensen, D., Vanishing theorems for complete intersections, J. Algebra 238 (2001), 684702.Google Scholar
[Jor08]Jorgensen, D., Finite projective dimension and the vanishing of Ext(M,M), Comm. Algebra 36 (2008), 44614471.Google Scholar
[Jot75]Jothilingam, P., A note on grade, Nagoya Math. J. 59 (1975), 149152.Google Scholar
[Lic66]Lichtenbaum, S., On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220226.CrossRefGoogle Scholar
[Mat86]Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986).Google Scholar
[Mil98]Miller, C., Complexity of tensor products of modules and a theorem of Huneke–Wiegand, Proc. Amer. Math. Soc. 126 (1998), 5360.Google Scholar
[MPSW10]Moore, F., Piepmeyer, G., Spiroff, S. and Walker, M., Hochster’s theta invariant and the Hodge–Riemann bilinear relations, Adv. Math. 226 (2010), 16921714.Google Scholar
[Rob76]Robbiano, L., Some properties of complete intersections in ‘good’ projective varieties, Nagoya Math. J. 61 (1976), 103111.Google Scholar
[Rob98]Roberts, P., Multiplicities and Chern classes in local algebra (Cambridge University Press, Cambridge, 1998).Google Scholar
[Ber04]Van den Bergh, M., Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel (Springer, Berlin, 2004), 749770.CrossRefGoogle Scholar