City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2005 Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings Mahdi Majidi-Zolbanin Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1765 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of NewYork. 2005 UMI Number: 3187456 Copyright 2005 by Majidi-Zolbanin, Mahdi All rights reserved. UMI Microform 3187456 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii c 2005 Mahdi Majidi-Zolbanin All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy. Lucien Szpiro Date Chair of Examining Committee Jozek Dodziuk Date Executive Officer Lucien Szpiro Raymond Hoobler Alphonse Vasquez Ian Morrison Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iv Abstract Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin Advisor: Professor Lucien Szpiro In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings. We give a splitting criterion for vector bundles of small i rank in terms of vanishing of their intermediate cohomology modules H (U, E )2≤i≤n−3, where n is the dimension of the regular local ring. This is the local analog of a result by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao for splitting of vector bundles of small rank on projective spaces. As an application we give a positive answer (in a special case) to a conjecture of R. Hartshorne asserting that certain quotients of regular local rings have to be complete intersections. More precisely we prove that if (R, m) is a regular local ring of dimension at least five, p is a prime ideal of codimension two, and the ring Γ(V, R/gp) is Gorenstein, where V is the open set Spec(R/p) − {m}, then R/p is a complete intersection. v To My Wife Leila vi Acknowledgments I would like to thank my advisor professor Lucien Szpiro, without whom this work would be impossible. It was an honor and a privilege to be his student, and benefit from his broad and profound mathematical knowledge and warm personality. My special thanks to professor Raymond Hoobler for reading this work, and for his valuable suggestions. I would also like to thank other members of my defense committee, professor Ian Morrison and Alphonse Vasquez. I am grateful to people at Mathematics Department of Graduate Center of City University of New York, specially professor Jozek Dodziuk for making my study at Graduate Center possible. I gained valuable teaching experience at Brooklyn College and Bronx Community College. I am thankful to professor George S. Shapiro and professor Roman Kossak for giving me the opportunity to work with them. Without my wife Leila, these long years would feel much longer. I am indebted to her for her continuous support and encouragement. This work is dedicated to her. vii The problem and its history: an epitome Let k be an algebraically closed field. One of the interesting problems in the area n of vector bundles on projective spaces Pk , and punctured spectrum of regular local rings, is the question of existence of indecomposable vector bundles of small rank on these spaces. Although such bundles exist in lower dimensions, they seem to become extremely rare as the dimension increases, especially in characteristic zero. In Table 1, n we have listed some known indecomposable vector bundles of small rank on Pk for n ≥ 4. Even though this list may not be complete, a complete list would not be much longer! As the table suggests, for n ≥ 6 it is not known whether there exist any indecom- n posable vector bundles E on Pk with 2 ≤ rank E ≤ n − 2. Also in characteristic zero 5 there are no known examples of indecomposable vector bundles of rank two on Pk. For vector bundles of rank two, there is the following Conjecture 0.1 (R. Hartshorne). [Har74, p. 1030] If n ≥ 7, there are no non-split n vector bundles of rank 2 on Pk . It is well-known that vanishing of certain (intermediate) cohomology modules of a vector bundle on projective space or punctured spectrum of a regular local ring, forces that bundle to split. Here are two important splitting criteria of this type: n Theorem 0.1 (G. Horrocks). If E is a vector bundle on Pk , then E splits, if and only viii n Table 1: Some indecomposable vector bundles of small rank on Pk , n ≥ 4. Constructed by Rank Base Space Characteristic Reference G. Horrocks, D. Mumford 2 P4 0 [HM73] C G. Horrocks 2 P4 2 [Hor80] H. Tango 2 P5 2 [Tan76b] N. Mohan Kumar 2 P4 p > 0 [Kum97] N. Mohan Kumar 2 P5 2 C. Peterson 2 P4 p > 0 [KPR02] A. P. Rao 3 P5 p > 0 3 P4 all H. Abo, W. Decker 3 P4 0 [ADS98] N. Sasakura G. Horrocks 3 P5 0, p > 2 [Hor78] H. Tango n − 1 Pn, n ≥ 3 all [Tan76a] U. Vetter n − 1 Pn, n ≥ 3 all [Vet73] i if H∗(E ) = 0, for 1 ≤ i ≤ n − 1. Theorem 0.2 (E. G. Evans, P. Griffith). [EG81, p. 331, Theorem 2.4] If E is a vector n i bundle on Pk of rank k < n, then E splits, if and only if H∗(E ) = 0, for 1 ≤ i ≤ k − 1. More recently, a new splitting criterion of this type was proved for vector bundles of n small rank on Pk by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao [KPR03, p. 185, Theorem 1]. For the statement of this result see Theorem II.2. In Theorem II.1, we extend this criterion to vector bundles of small rank on punctured spectrum of regular local rings. The proof is an adapted version of the original proof, that suits the setting of local rings. Chapter II is devoted to the proof of this theorem. Another interesting question, related to the vector bundle problem, is the question n of existence of non-singular subvarieties of Pk (k algebraically closed) of small codi- mension, which are not complete intersections. Here is the precise statement of this problem: ix Conjecture 0.2 (R. Hartshorne). [Har74, p. 1017] If Y is a nonsingular subvariety n 2 of dimension r of Pk , and if r > 3 n, then Y is a complete intersection. Conjecture 0.2 has also a local version, which is in fact stronger than the original conjecture. To see the motivation behind the local conjecture, let Y be a non-singular n subvariety of dimension r in Pk , defined by a homogeneous prime ideal p of the poly- n+1 nomial ring S := k[X0, ··· ,Xn]. Let R be the local ring of the origin in Ak , that is, R = Sm, where m is the maximal ideal (X0, ··· ,Xn) in S. Then Y is a complete n intersection in Pk , if and only if R/pR is a complete intersection in R. Furthermore, R is a regular local ring of dimension n + 1, the quotient ring R/pR has an isolated 2 singularity, and dim R/pR = r + 1. The inequality r > 3 n of conjecture 0.2 can be 2 rewritten in terms of dim R/pR and dim R, as dim R/pR − 1 > 3 (dim R − 1), or after 1 simplifying, dim R/pR > 3 (2 dim R + 1). The local version of Conjecture 0.2 is the following: Conjecture 0.3 (R. Hartshorne). [Har74, p. 1027] Let (R, m) be a regular local ring with dim R = n, let p ⊂ R be a prime ideal, such that R/p has an isolated singularity, 1 let r = dim R/p, and suppose that r > 3 (2n + 1). Then R/p is a complete intersection. There are some affirmative answers to Conjecture 0.3, obtained under additional hypotheses on depth(m, R/p). We mention some of these results in codimension 2, i.e., when dim R/p = dim R − 2: Theorem 0.3 (C. Peskine, L. Szpiro). [PS74, p. 294, Theorem 5.2] Let R be a regular local ring with dim R ≥ 7. Let R/a be a quotient of codimension 2 of R, which is Cohen-Macaulay, and locally a complete intersection except at the closed point. Then R/a is a complete intersection. x Theorem 0.4 (R. Hartshorne, A. Ogus). [HO74, p. 431, Corollary 3.4] Let (R, m) be a regular local ring containing its residue field k of characteristic 0. Let p ⊂ R be a prime ideal of codimension 2 in R, such that R/p is locally a complete intersection 1 except at the closed point. Assume that n = dim R ≥ 7, and depth(m, R/p) > 2 (r + 1), where r = dim R/p.