Bounds for Codimensions of Fitting Ideals

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1 Ž. JOUNAL OF ALGEBA 194, ATICLE NO. JA Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, , Kakow, Poland Communicated by Caig Huneke eceived June 25, 1996 We find uppe bounds fo coensions of Fitting ideals of a module Žove a egula ing., whose symmetic algeba is equiensional. These bounds ae stonge than the classical Eagon Nothcott bounds. Togethe with lowe bounds of Simis and Vasconcelos, they show that the Fitting ideals of a module whose symmetic algeba has an ieducible spectum can appea only in a finite numbe of coensions. Ou poof is based on ealie esults concening minimal pimes of tenso powes of a symmetic algeba and uses one of the ension fomulæ of Huneke and ossi Academic Pess Thoughout the pape, shall be a commutative egula Noetheian domain which is a finitely geneated k-algeba, ove a field k of zeo chaacteistic. Futhe, M shall be a finitely geneated -module. The symmetic algeba of M shall be denoted Sym M. The main esult of the pape is the following: THEOEM 1. Fo a fixed 1, let Q be a minimal associated pime of Fitt 1M, not being an associated pime of Fitt M. If Spec Sym M is equiensional Ž as a scheme o e Spec k., then co Q Ž k M.. Let us stess that by equiensionality, we mean absolute equiensionality and not equiensionality as a scheme ove Spec, which would imply that M is pojective and the bound would be tivial. Futhe, *Wok patially suppoted by KBN Gant 2-P03A The autho thanks Jean-Paul Basselet and the Institut de Mathematiques de Luminy Žwhee pat of this wok was caied out., fo poviding excellent woking conditions in the sping of addess: kwiecins@im.uj.edu.pl $25.00 Copyight 1997 by Academic Pess All ights of epoduction in any fom eseved. 378

2 FITTING IDEALS 379 notice that the assumptions of ou theoem do not imply that Q is a contaction of a minimal pime of Sym M. emak also, that if we take Q to be any minimal associated pime of any Fitting ideal of M and set k Q M Q M then the assumptions of ou theoem will be satisfied fo and Q Q. Theefoe, the theoem povides infomation about the coensions of all minimal associated pimes of all Fitting ideals. Fom a staightfowad computation of the ensions of fibes of the map Spec Sym M Spec, Ž 1. one deduces the following lowe bounds: if Spec Sym M is equiensional, then co Q k M, and if Spec Sym M is ieducible, then co Q k M 1. Ž 2. Ž. This last bound is due to Simis and Vasconcelos see 10, 11. Togethe with ou bound, these esults show that co Q can take only a finite numbe of values. In paticula: COOLLAY 2. co Q 2. If Spec Sym M is ieducible, km 1and 2 then Anothe eason fo having a necessay condition fo the equiensionality of Spec Sym M is that ieducibility is equivalent to equiensionality plus the condition 2, known as condition F Žsee Now, fo any m n matix A, with enties in, the coension of the ideal I of t-minos of A, satisfies the Eagon Nothcott inequality Ž t see 3, 4. : co It Ž m t 1.Ž n t 1., povided that It. Fo M coke A, ou bound would in this case be Ž m t 1.Ž k A t 1., whee k A is the geneic ank of the matix. It is theefoe obvious, that fo modules with pojective ension geate than one, ou bounds ae stonge than the Eagon Nothcott ones. An example of this situation is k x, y, z,..., z, and 1 s x n x n 1 y xy n 1 A. n 1 n 2 2 n x y x y y

3 380 MICHAŁ KWIECINSKI Then Spec Sym M is ieducible Ž although not educed. and co Fitt 1 M 2, so the bound in Theoem 1 is shap, wheeas the Eagon Nothcott bound would give 2n. In the above example ou bound is also shape than the Eisenbud Evans Buns bound fo t with It 1 0, co I m n 2t 1 t Žconjectued in 5 and poved in 2, Theoem 2., and the Buns bound 2, Theoem 3, also with I 0, t 1 co It maxž n, m t 1.. Poof of Theoem 1. Theoem 1 will esult fom a study of tenso powes of the symmetic algeba of M and a computation of ensions of thei minimal pimes. We will use a theoem fom 8, which elates such pimes to the Fitting ideals of M. Conside the tenso powe Sym M Ž the tenso poduct of copies of Sym M ove.. It is of couse isomophic to SymŽM.. As in 8, Definition 5, let M be the family of those minimal pimes P of Sym M, such that ž / Sym M P Ž P. 2. Ž 3. By 8, Definition 5 and Theoem 7, we have ' Fitt M P. Ž 4. 1 P M Localize at an element not belonging to Q, but belonging to all othe minimal pimes of Fitt 1 M. We now have ' Fitt 1 M Q. This implies that the contaction of the intesection of pimes in 4 is a pime and theefoe fo some P1 M, Q P 1. Ž 5. By the Huneke ossi ension fomula Ž 6, Theoem 2.6.i. applied to SymŽ M.., the diffeence of ensions in Ž 3. is equal to the ank: k Ž P. M Ž P. M. 2 But, fo P P 1, by 5 and the assumptions on Q, this ank is equal to. Hence 1 Sym M P Q 2. Ž 6.

4 FITTING IDEALS 381 Ž Now, we shall compute Sym M. P1 in a diffeent way, which shall lead to the bound in ou theoem. Conside the diagonal ideal, kenel of the multiplication map: ke. We have the following obvious isomophism of k-algebas: k Sym M Sym M Sym M. k k Since Sym M is equiensional, so is S Sym M Žby k 7, Kapitel II, Koolla 3.9.b. and S Ž k M.. Now is the ideal of the diagonal embedding of Spec in the fibed poduct of copies of Spec ove Spec k. Since the field k is of zeo chaacteistic Ž and hence pefect. and is egula, Spec is smooth Žby 1, VII Thus is the ideal of a smooth subscheme of a smooth scheme. Hence Žby 1, VII.5.8 and VII.5.9., in any localization S Ž m whee m is any maximal ideal in S, containing. the ideal S has a system of Ž 1. m geneatos. Now P1 is a minimal pime of S S. Because S is equiensional, we can apply Kull s Pincipal Ideal Theoem and obtain 1 Ž k M. Sym M P Ž 1.. Ž 7. The above inequality, togethe with 6 gives the bound of Theoem 1, ending the poof of that theoem. The eade might have noticed that the geometic idea behind this poof is elated to studying finite sets in the fibes of the map 1 that lie in the limits of sequences of geneic fibes 9. To conclude, we state a shapened, but less elegant vesion of Theoem 1. THEOEM 3. Q 0 and let then If, in addition to the assumptions of Theoem 1, we suppose s supk IM IM: I Spec, I Q, I Q 4, co Q Ž s 1.Ž k M.. To pove this, one should just adapt the poof of Theoem 1, this time consideing the family of those minimal pimes P of s 1 Sym M, such that ž / s 1 Sym M P Ž P. Ž s 1..

5 382 MICHAŁ KWIECINSKI Then, afte a suitable localization, an analogue of fomula 4 will hold. The est of the poof caies ove with no majo diffeence. EFEENCES 1. A. Altman and S. Kleiman, Intoduction to Gothendieck duality theoy, in Lectue Notes in Math., Vol. 146, Spinge-Velag, New Yok Belin, W. Buns, The Eisenbud Evans genealized pincipal ideal theoem and deteminantal ideals, Poc. Ame. Math. Soc. 83 Ž 1981., A. Eagon and D. J. Nothcott, Ideals defined by matices and a cetain complex associated with them, Poc. oy. Soc. London Se. A 269 Ž 1962., D. Eisenbud, Commutative algeba with a view towad algebaic geomety, in Gaduate Texts in Math., Vol. 150, Spinge-Velag, New Yok Belin, D. Eisenbud and E. G. Evans, J., A genealized pincipal ideal theoem, Nagoya Math. J. 62 Ž 1976., C. Huneke and M. ossi, The ension and components of symmetic algebas, J. Algeba 98 Ž 1986., E. Kunz, Einfuhung in die kommutative Algeba und algebaische Geometie, in Vieweg Stud., Vol. 46, Vieweg, Wiesbaden, M. Kwiecinski, Tenso powes of symmetic algebas, Comm. Algeba 24 Ž 1996., M. Kwiecinski and P. Twozewski, Finite sets in fibes of holomophic maps, pepint, IMUJ , Jagiellonian Univesity, Kakow. 10 A. Simis and W. V. Vasconcelos, Kull ension and integality of symmetic algebas, Manuscipta Math. 61 Ž 1988., W. V. Vasconcelos, Aithmetic of blowup algebas, in London Math Soc. Lectue Note Se., Vol. 195, Cambidge Univ. Pess, Cambidge, UK, 1994.

SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES

italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics