# One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Open Mathematics (2011)

- Volume: 9, Issue: 6, page 1349-1353
- ISSN: 2391-5455

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topVeronique Lierde. "One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal." Open Mathematics 9.6 (2011): 1349-1353. <http://eudml.org/doc/268961>.

@article{VeroniqueLierde2011,

abstract = {Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.},

author = {Veronique Lierde},

journal = {Open Mathematics},

keywords = {Degree function; Rees valuation; 2-dimensional rational singularity},

language = {eng},

number = {6},

pages = {1349-1353},

title = {One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal},

url = {http://eudml.org/doc/268961},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Veronique Lierde

TI - One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

JO - Open Mathematics

PY - 2011

VL - 9

IS - 6

SP - 1349

EP - 1353

AB - Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.

LA - eng

KW - Degree function; Rees valuation; 2-dimensional rational singularity

UR - http://eudml.org/doc/268961

ER -

## References

top- [1] Debremaeker R., First neighborhood complete ideals in two-dimensional Muhly local domains, J. Pure Appl. Algebra, 2009, 213(6), 1140–1151 http://dx.doi.org/10.1016/j.jpaa.2008.11.002 Zbl1163.13004
- [2] Debremaeker R., Van Lierde V., The effect of quadratic transformations on degree functions, Beiträge Algebra Geom., 2006, 47(1), 121–135 Zbl1095.13026
- [3] Debremaeker R., Van Lierde V., On adjacent ideals in two-dimensional rational singularities, Comm. Algebra, 2010, 38(1), 308–331 http://dx.doi.org/10.1080/00927870903392476 Zbl1201.13007
- [4] Göhner H., Semifactoriality and Muhly’s condition (N) in two dimensional local rings, J. Algebra, 1975, 34(3), 403–429 http://dx.doi.org/10.1016/0021-8693(75)90166-0
- [5] Huneke C., Complete ideals in two-dimensional regular local rings, In: Commutative Algebra, Berkeley, June 15–July 2, 1987, Math. Sci. Res. Inst. Publ., 15, New York, Springer, 1989, 325–338
- [6] Huneke C., Swanson I., Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser., 336, Cambridge University Press, Cambridge, 2006 Zbl1117.13001
- [7] Lipman J., Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 1969, 36, 195–279 http://dx.doi.org/10.1007/BF02684604 Zbl0181.48903
- [8] Noh S., Simple complete ideals in two-dimensional regular local rings, Comm. Algebra, 1997, 25(5), 1563–1572 http://dx.doi.org/10.1080/00927879708825936 Zbl0878.13015
- [9] Noh S., Watanabe K., Adjacent integrally closed ideals in 2-dimensional regular local rings, J. Algebra, 2006, 302(1), 156–166 http://dx.doi.org/10.1016/j.jalgebra.2005.10.034 Zbl1109.13019
- [10] Rees D., Degree functions in local rings, Math. Proc. Cambridge Philos. Soc., 1961, 57(1), 1–7 http://dx.doi.org/10.1017/S0305004100034794 Zbl0111.24901
- [11] Rees D., Sharp R.Y., On a theorem of B. Teissier on multiplicities of ideals in local rings, J. Lond. Math. Soc., 1978, 18(3), 449–463 http://dx.doi.org/10.1112/jlms/s2-18.3.449 Zbl0408.13009
- [12] Van Lierde V., A mixed multiplicity formula for complete ideals in 2-dimensional rational singularities Proc. Amer. Math. Soc., 2010, 138(12), 4197–4204 http://dx.doi.org/10.1090/S0002-9939-2010-10455-0 Zbl1207.13006
- [13] Van Lierde V., Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal, J. Pure Appl. Algebra, 2010, 214(5), 512–518 http://dx.doi.org/10.1016/j.jpaa.2009.06.002 Zbl1184.13026
- [14] Zariski O., Polynomial ideals defined by infinitely near base points, Amer. J. Math., 1938, 60(1), 151–204 http://dx.doi.org/10.2307/2371550 Zbl0018.20101
- [15] Zariski O., Samuel P., Commutative Algebra. II, The University Series in Higher Mathematics, Van Nostrand, Princeton-Toronto-London-New York, 1960 Zbl0121.27801

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