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

Veronique Lierde

Open Mathematics (2011)

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

Abstract

top
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.

How to cite

top

Veronique 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] Zariski O., Samuel P., Commutative Algebra. II, The University Series in Higher Mathematics, Van Nostrand, Princeton-Toronto-London-New York, 1960 Zbl0121.27801

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.