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An intrinsic definition of the Rees algebra of a module

Part of: General commutative ring theory Theory of modules and ideals

Published online by Cambridge University Press:  22 February 2017

Gustav Sædén Ståhl
Gustav Sædén Ståhl*
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden ([email protected])
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Abstract

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

Keywords

Rees algebrasdivided powersbialgebras

MSC classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C12: Torsion modules and ideals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 13 - 30
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1. Aluffi, P., Shadows of blow-up algebras, Tohoku Math. J. 56(4) (2004), 593–619.Google Scholar
2. Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, Volume 39 (Cambridge University Press, 1998).CrossRefGoogle Scholar
3. Eisenbud, D., Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, Volume 150 (Springer, 1995).CrossRefGoogle Scholar
4. Eisenbud, D., Huneke, C. and Ulrich, B., What is the Rees algebra of a module?, Proc. Amer. Math. Soc. 131(3) (2003), 701–708.Google Scholar
5. Gaffney, T. and Kleiman, S. L., Specialization of integral dependence for modules, Invent. Math. 137(3) (1999), 541–574.Google Scholar
6. Katz, D., Reduction criteria for modules, Commun. Alg. 23(12) (1995), 4543–4548.Google Scholar
7. Katz, D. and Kodiyalam, V., Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Am. Math. Soc. 349(2) (1997), 747–762.Google Scholar
8. Kleiman, S. and Thorup, A., Conormal geometry of maximal minors, J. Alg. 230(1) (2000), 204–221.Google Scholar
9. Kodiyalam, V., Integrally closed modules over two-dimensional regular local rings, Trans. Am. Math. Soc. 347(9) (1995), 3551–3573.Google Scholar
10. Liu, J.-C., Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring, Commun. Alg. 26(12) (1998), 4015–4039.Google Scholar
11. Rees, D., Reduction of modules, Math. Proc. Camb. Phil. Soc. 101(3) (1987), 431–449.Google Scholar
12. Roby, N., Lois polynomes et lois formelles en théorie des modules, Annales scient. Éc. Norm. Sup. 80(3) (1963), 213–348.Google Scholar
13. Rydh, D., Families of cycles and the Chow scheme, PhD thesis, KTH Kungliga Tekniska högskolan (2008).Google Scholar
14. Ståhl, G. Sædén, Rees algebras of modules and coherent functors, Preprint (arXiv:1409.6464; 2014).Google Scholar
15. Ståhl, G. Sædén, Total blow-ups of modules and universal flatifications, Commun. Alg. (2016), DOI:10.1080/00927872.2016.1244270.Google Scholar
16. Simis, A., Ulrich, B. and Vasconcelos, W. V., Rees algebras of modules, Proc. Lond. Math. Soc. 87 (1999), 610–646.Google Scholar
17. Simis, A., Ulrich, B. and Vasconcelos, W. V., Codimension, multiplicity and integral extensions, Math. Proc. Camb. Phil. Soc. 130(2) (2001), 237–257.CrossRefGoogle Scholar
18. Vasconcelos, W. V., Arithmetic of blowup algebras, London Mathematical Society Lecture Notes, Volume 195 (Cambridge University Press, 1994).CrossRefGoogle Scholar