Journal of Algebra. A class of projectively full ideals in two-dimensional Muhly local domains

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1 Journal of Algebra Contents lists available at ScienceDirect Journal of Algebra wwwelseviercom/locate/jalgebra A class of projectively full ieals in two-imensional Muhly local omains aymon Debremaeker Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 300 Leuven, Belgium article info abstract Article history: eceive 5 June 2008 Availableonline7November2008 Communicate by Luchezar L Avramov Keywors: Integrally close ieal Projectively equivalent ieals Projectively full ieal eesvaluationofanieal Let, M be a regular local omain of imension 2 an let,, be a regular system of parameters Then, Ciuperca, Heinzer, atliff Jr, an ush have prove that every ieal of the form I = n,, with n N +, is projectively full If, M is a two-imensional Muhly local omain ie, an integrally close Noetherian local omain with algebraically close resiue fiel an the associate grae ring an integrally close omain, then we are able to prove a similar result for every minimal ieal basis,, of M such that / ra 2,, 2008 Elsevier Inc All rights reserve Introuction Let I be a regular proper ieal of a Noetherian ring ie, I contains an element with zero annihilator an I An ieal J of is projectively equivalent to I if there eist positive integers n an m such that I n an J m have the same integral closure, ie, I n = J m Here the overbar - is use to enote the integral closure of an ieal In [2, Ciuperca, Heinzer, atliff Jr, an ush, introuce the notion of projectively full ieal An ieal I as above is calle projectively full if the set PI of integrally close ieals projectively equivalent to I is the set {I n n N + } In Proposition 36 of [3 Ciuperca, Heinzer, atliff Jr, an ush prove the following: Let, M be a local ring an I a normal ieal in with I M 2 If both the associate grae ring GM = M n n 0 M an the fiber cone ring F I = I n n 0 MI n are reuce, then I is projectively full This proposition implies the following result [3, Corollary 37: aress: raymonebremaeker@wiskuleuvenbe /$ see front matter 2008 Elsevier Inc All rights reserve oi:006/jjalgebra

2 904 Debremaeker / Journal of Algebra Let, M be a regular local omain of altitue 2 an let,, be a regular system of parameters Then every ieal of the form I = n,, is projectively full The purpose of this paper is to prove a similar result in case, M is a two-imensional Muhly local omain, ie, a two-imensional normal Noetherian local omain with algebraically close resiue fiel an the associate grae ring an integrally close omain More precisely we will prove the following Proposition Let, M be a two-imensional Muhly local omain an let,, be a minimal ieal basis of M such that / ra 2,, Then every ieal of the form I = n,, with n N +,is projectively full In Section 2 we will prove that the ieals of the form I = n,, are contracte from, ie, [ M I = I Using this, it will follow that these ieals an their transforms for eplanation of terminology, see Section 2 are integrally close or complete for all n These facts together with the observation that the blowup Bl M of at M is a esingularization of, will enable us to eploit Zariski s theory of complete ieals in two-imensional regular local rings to prove our main result in Section 3 For the efinition an properties of the blowup of a ring at an ieal, the reaer is referre to [6 Backgroun information on Zariski s theory of complete ieals in 2-imensional regular local rings can be obtaine from [7, [, Chapter 4 an [2 Goo references for briefly state efinitions an theorems from Zariski s theory of complete ieals are [9 an [0 2 Preliminaries Let, M be a two-imensional Muhly local omain an let,, be a minimal ieal basis of M such that / ra 2,, Note that the conition / ra 2,, isequivalent to 2,, is a prime ieal of an also equivalent to 2,, is a one-imensional regular local ring, an it implies that 2 = 2,, It follows that M := is a maimal ieal of lying over M ie, M = M an the ring [ M := is a two-imensional regular local ring To see this, note that the associate grae ring of being a two-imensional integrally close omain, implies that the close fibre of the blowup of at M is a non-singular curve It follows that every local ring of Bl M is regular see [5, p 403 an [8, p 259 The local ring is calle an immeiate or a first quaratic transform of Let I := n,, with n 2 Then in the ring, one has [ M I = M n

3 Debremaeker / Journal of Algebra The ieal J := n is calle the transform of I in It is clear that M is the only prime ieal of containing J Localizing at M yiels in I = J M an the ieal I := J M is calle the transform of I in, M, where M enotes the maimal ieal of Since the transform I, the immeiate quaratic transform, M of, M is calle an immeiate base point of I Fori {2,,}, wehave [ M I = i i i, hence there are no immeiate base points of I on the chart This shows that i, M is in fact the only immeiate base point of I We are now reay to prove a few results about the ieals I = n,, which will be use in the proof of our main result Lemma 2 Every ieal of the form I = n,, with n is contracte from Proof We procee by inuction on n an we first consier the case n = Then it is clear that I = I because I = M in that case Now suppose that n,, = n,, Then we have to prove that,, =,, To this en we observe that,,,, n,, Now,, n,, are ajacent ieals, ie, their lengths iffer by one To see this it suffices to observe that n /,, because / ra 2,, It follows from the ajacentness that,, =,, or n,, So it remains to prove that,, = n,, is not possible For if this woul be the case, then one woul have n,, =,,, hence n = n

4 906 Debremaeker / Journal of Algebra an thus n = n This woul imply that in the one-imensional regular local ring S := 2,, one woul have N n = N n, where N enotes the maimal ieal of S This is impossible an hence,, =,, This result will be use to prove in the net lemma that the ieals I = n,, are complete Lemma 22 With the previous assumptions an notations, we have: i Every ieal n of is complete ii Every ieal n,, of is complete Proof First let us recall that 2 = 2,, since 2 = 2,,,anthering S = 2,, is a one-imensional regular local omain with maimal ieal N = S, where enotes the natural image of in S Let J := n J, then the ieal 2,, of S is equal to N n an hence integrally close This implies that J is also integrally close, which proves i To prove ii, we may suppose n 2, the case n = beingtrivialin we have that [ n, M 2,, = n Since, M is a two-imensional Muhly local omain, the ring is an integrally close omain an since n is a complete ieal of, it follows that n is complete too But [ n, M 2,, = n, 2,, by Lemma 2 an thus n,, is complete Corollary 23 Every ieal of the form I = n,,, n 2, has just one immeiate base point, M where = M with M := an the transform I of I in is a simple complete ieal Proof It is clear that I = i i i for i = 2,,, hence the immeiate base points of I can only show up on the chart In the ring we have [ M I = n where J := n is the transform of I in an M is the only prime ieal of containing J It follows that M is the only immeiate base point of I

5 Debremaeker / Journal of Algebra The transform J of I in is complete because of Lemma 22i This implies that the transform I = J M of I in is also complete, an I is simple ie, not a prouct of two proper ieals because or I = In orer to state our net result we recall that, M being a two-imensional Muhly local omain implies that the M-aic orer function or is a valuation which will be enote by v M If I = n,, with n 2, then I has only one immeiate base point, M see Corollary 23, an is a two-imensional regular local ring Since the transform I of I in the two-imensional regular local ring is a simple complete ieal, we know by Zariski s theory of complete ieals that there correspons to I a unique prime ivisor of, namely the unique ees valuation of I,enote by w see eg [4, Corollaries 36 an 37 Thus T I ={w} Throughout this paper the set of ees valuations of an ieal A in a local ring will be enote by T A Lemma 24 For every ieal of the form I = n,, with n 2, wehavethatti {v M, w} an w T I Proof Since, M is the only immeiate base point of I, it follows that the blowup Bl IM is obtaine by blowing up at M an then blowing up the local ring, M Bl M at the transform I of I in see [6, Lemma This implies that T IM ={v M, w} an hence T I {v M, w} because T IM = T I T M cf [, Proposition 048 Finally we have to show that w T I Suppose not, then one woul have that T I ={v M This woul imply that I has no immeiate base points, contraicting Corollary 23 We close this section by recalling information on the transform I of I = n,, n 2 in its unique immeiate base point, M an on the unique ees valuation w of I Sincew is a prime ivisor of ie, the valuation ring W, M W of w ominates, M an the transcenence egree of the resiue fiel of W over M is one, we know by Abhyankar [, Proposition 3 that there eists a unique finite quaratic sequence starting from an ominate by W, M W :, M =, M < 2, M 2 < < s, M s <W, M W, ie, i, M i is an immeiate quaratic transform of i, M i for i = 2,,s an W is the or s -valuation ring Thus the unique ees valuation w of I is the or s -valuation Moreover the transform of I in s,enotei s, is the maimal ieal of s, ie, I s = M s, an conversely, the inverse transform of M s in is I see eg [9, p Proof of the main result Let, M be a two-imensional Muhly local omain an let,, be a minimal ieal basis of M such that / ra 2,, Then we have to prove that the ieal I := n,, is projectively full for all integers n The case n = is clear since M has only one ees valuation, namely v M = or -valuation So we may assume in the rest of the proof that n 2 Let J be a complete M-primary ieal of that is projectively equivalent to I ie, I i = J j with i, j positive integers Then we have to prove that J = I ν for some positive integer ν As J is projectively equivalent to I, J has the same ees valuations as I, hencet J = T I

6 908 Debremaeker / Journal of Algebra Since T I {v M, w} an w T I see Lemma 24, it follows that J,justlikeI, has precisely one immeiate base point, namely the center of w on Bl M, ie,, M Inee, it is clear that J has at least one immeiate base point since otherwise one woul have that Bl J is ominate by Bl M an thus T J T M ={v M } by Proposition 22 in [5 This woul imply that T I ={v M } contraicting Lemma 24 Any immeiate base point of J is ominate by a ees valuation ring of J one can prove this using an argument similar to that use by Göhner in his proof of Proposition 22 in [5, hence by the valuation ring W, M W of w since w is the only element v M in T J This proves that J has only one immeiate base point, namely the unique local ring of Bl M ominate by W, M W which is calle the center of w on Bl M It follows from the iscussion at the en of Section 2 that the center of w on Bl M is, M Hence,, M istheuniqueimmeiatebasepointof J Net, we prove that w is the only ees valuation of the integral closure J of the transform J of J in its unique immeiate base point, M Letr:= or J, thenwehavein that J = r J an the ieal J is the transform of J in, M Note,weonotknowwhether J is complete or not It follows that J = r J where J is a complete M -primary ieal in the two-imensional regular local ring, M an we claim that T J ={w} In orer to prove the claim, let us suppose that there eists a ees valuation w of J with w w We now show this leas to a contraiction Since w is a ees valuation of J,wehave W Bl J where W enotes the valuation ring of w Because J = r J,wehaveBl J = Bl J see [6 From the fact that, M is the only immeiate base point of J, it follows that the blowup Bl JM is obtaine by blowing up at M an then blowing up the local ring, M Bl M at J see [6, Lemma This implies that W Bl JM an it follows that w T JM = T M T J where T M ={v M } an T J {v M, w} As w w, wehavew = v M, which implies that W Bl M SinceW ominates the local ring Bl M, we have the esire contraiction Thus J cannot have ees valuations w an this proves that T J ={w}, since every ieal in a Noetherian ring has a set of ees valuations see [, Theorem 022 From the theory of complete ieals in a two-imensional regular local ring see for eample [4,9,0, we know that J is some power of the simple complete M -primary ieal that correspons to w via Zariski s one-to-one corresponence This unique simple complete M -primary ieal is the inverse transform of M s in, where M s is as in the iscussion at the en of Section 2 cf [, Theorem 022 Since we know that M s is the transform of I in s, we have conversely that I = n is the inverse transform of M s in see the en of Section 2 Thus J = I ν = n ν for some ν N +

7 Debremaeker / Journal of Algebra The fact that J is projectively equivalent to I will enable us to prove that ν = r, where r = or J Since I i = J j for some i, j N +,or I = an or J = r, it follows that i = rj an hence J j = I rj ecalling that I = n, we have that J j = rj n rj On the other han, J = r J implies Comparing an 2 yiels J j = J j = rj J j = rj n ν j 2 n rj = n ν j It follows that rj = ν j, hencer = ν Thus J = I r = n r an hence J = r J = r I r = I r = I r From J = I r, it follows that J = I r To see this, consier the following sequence of inclusions J J JV M JW, where V M is the valuation ring of v M = or an W is the valuation ring of w Since T J {v M, w} an J is complete, we have an thus JV M JW = J J = J

8 90 Debremaeker / Journal of Algebra Since J = I r, we see that J = I r in particular I r is a complete ieal of Finally we prove that I r = I r SinceI r I r an I r is complete, we have that I r I r Further we have the following inclusions an I r I r I r V M I r W, I r V M I r W = I r because T I r = T I {v M, w} an w T I r,hence I r = I r Thus J = I r, ie, I = n,, is projectively full Final emark In the special case where, M is a two-imensional regular local ring, any minimal ieal basis of M satisfies the conition / ra 2 However, this is not necessarily true if, M is a two-imensional Muhly local omain that is not regular, as the eample below shows Let := k[x, Y, Z X,Y,Z X 2 YZ X,Y,Z, with k an algebraically close fiel an X, Y, Z ineterminates over k It follows that = k[, y, z,y,z with 2 = yz, where, y, z enote the natural images of X, Y, Z Then, y,, z is a minimal ieal basis of M such that y / ra, z, while the minimal ieal basis, y, z of M clearly oes not satisfy the conition / ray, z More generally, let, M be any two-imensional Muhly local omain of the form := k[x,,x X,,X, N X,,X where k is an algebraically close fiel, X,,X are ineterminates over k, an N is a homogeneous prime ieal of height 2ink[X,,X Using the projective Nullstellensatz, it can be shown that there eists a minimal ieal basis, 2,, of M, such that 2,, is a prime ieal Hence / ra 2,, eferences [ S Abhyankar, On the valuations centere in a local omain, Amer J Math [2 C Ciuperca, WJ Heinzer, LJ atliff Jr, DE ush, Projectively equivalent ieals an ees valuations, J Algebra [3 C Ciuperca, WJ Heinzer, LJ atliff Jr, DE ush, Projectively full ieals in Noetherian rings, J Algebra [4 Debremaeker, V Van Liere, The effect of quaratic transformations on egree functions, Beitrage Algebra Geom [5 H Göhner, Semifactoriality an Muhly s conition N in two imensional local rings, J Algebra [6 W Heinzer, B Johnston, D Lantz, K Shah, Coefficient ieals in an blowups of a commutative Noetherian omain, J Algebra

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