TY - JOUR
T1 - Stable division and essential normality
T2 - the non-homogeneous and quasi-homogeneous cases
AU - Biswas, Shibananda
AU - Shalit, Orr Moshe
N1 - Funding Information:
The first author’s work is partially supported by an Inspire Faculty Fellowship (IFA-11MA-06) funded by DST, India. The second author’s work is partially supported by an ISF grant (no. 474/12), and by an EU FP7/2007-2013 grant (no3˙21749).
Funding Information:
The authors are grateful to an anonymous referee for spotting a couple of mistakes in a previous version of this paper, and for several other helpful remarks that improved our presentation. The authors also wish to thank Guy Salomon for providing useful feedback. The first author’s work is partially supported by an Inspire Faculty Fellowship (IFA-11MA-06) funded by DST, India. The second author’s work is partially supported by an ISF grant (no. 474/12), and by an EU FP7/2007-2013 grant (no321749).
Publisher Copyright:
© 2018 Department of Mathematics, Indiana University. All rights reserved.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - Let H d (t) (t ≥ −d, t > −3) be the reproducing kernel Hilbert space on the unit ball B d with kernel 1 k(z, w) = (1 − h z, wi )d +t+ 1 . We prove that if an ideal I C[z 1 , . . ., z d ] (not necessarily homogeneous) has what we call the approximate stable division property, then the closure of I in H d (t) is p-essentially normal for all p > d. We then show that all quasi-homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi-homogeneous ideal in C[x, y] is p-essentially normal for p > 2.
AB - Let H d (t) (t ≥ −d, t > −3) be the reproducing kernel Hilbert space on the unit ball B d with kernel 1 k(z, w) = (1 − h z, wi )d +t+ 1 . We prove that if an ideal I C[z 1 , . . ., z d ] (not necessarily homogeneous) has what we call the approximate stable division property, then the closure of I in H d (t) is p-essentially normal for all p > d. We then show that all quasi-homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi-homogeneous ideal in C[x, y] is p-essentially normal for p > 2.
KW - Essential normality
KW - Groebner basis
KW - Hilbert submodules
KW - Quasi-homogeneous polynomials
UR - http://www.scopus.com/inward/record.url?scp=85064276894&partnerID=8YFLogxK
U2 - 10.1512/iumj.2018.67.6272
DO - 10.1512/iumj.2018.67.6272
M3 - Article
AN - SCOPUS:85064276894
SN - 0022-2518
VL - 67
SP - 169
EP - 185
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 1
ER -