Hilbert Functions, Residual Intersections, and Residually S 2 Ideals

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1 Composito Mathematica 125: 193^219, # 2001 Kluwe Academic Publishes. Pinted in the Nethelands. Hilbet Functions, Residual Intesections, and Residually S 2 Ideals MARC CHARDIN 1, DAVID EISENBUD 2 * and BERND ULRICH 3 * 1 Institut de Mathe matiques, CNRS Univesite Pais 6, 4, place Jussieu, F^75252 Pais Cedex 05, Fance. chadin@math.jussieu.f 2 Mathematical Sciences Reseach Institute, 1000 Centennial Dive, Bekeley, CA 94720, U.S.A. de@msi.og 3 Depatment of Mathematics, Michigan State Univesity, East Lansing, MI 48824, U.S.A. ulich@math.msu.edu (Received: 21 Decembe 1998; accepted in nal fom: 15 Novembe 1999) Abstact. Let R be a homogeneous ing ove an in nite eld, I R a homogeneous ideal, and a I an ideal geneated by s foms of degees d 1 ;...; d s so that codim a : I X s.we give boad conditions fo when the Hilbet function of R=a o of R= a : I is detemined by I and the degees d 1 ;...; d s. These conditions ae expessed in tems of esidual intesections of I, culminating in the notion of esidually S 2 ideals. We pove that the esidually S 2 popety is implied by the vanishing of cetain Ext modules and deduce that geneic pojections tend to poduce ideals with this popety. Mathematics Subject Classi cations (2000). Pimay: 13D40, 13C40; Seconday: 13H15, 14M06. Key wods. Hilbet function, Hilbet polynomial, esidual intesection, esidually S 2, pasimonious, thifty. Intoduction Let I be a homogeneous ideal in a polynomial ing R ˆ k x 1 ;...; x n Š ove an in nite eld k. Choose degees d 1 ;...; d s, and conside the family of ideals a geneated by elements a 1 ;...; a s 2 I of degees d 1 ;...; d s. By semicontinuity thee ae open sets in this family consisting of ideals a such that the Hilbet functions of R=a and of the esidual intesection R= a : I ae constant. One might ask: (A) What do these open sets look like? (B) What ae these geneic Hilbet functions? If I ˆ x 1 ;...; x n and s W n, then the answes ae well known: The open sets of question (A) each consist pecisely of the ideals a of codimension s, and the geneic Hilbet seies of question (B) ae Q 1 t d i = 1 t n except in case s ˆ n, whee *Patially suppoted by the NSF.

2 194 MARC CHARDIN ET AL. R= a : I is R=a modulo its socle, and has Hilbet seies Q 1 t d i P 1 t n t di 1 : Even fo this I, both questions ae open when s > n (see [24], [9], [1] and [13] fo patial esults). In this pape we will take up question (A), always in the case s W n, but fo moe geneal ideals I. Ou main esults ae that, in a wide ange of cases, codimension conditions de ne the open sets in question. In a late pape [8] we will compute the geneic Hilbet functions of question (B) unde stonge hypotheses on I. Fo the pupose of descibing ou esults, we say that the ideal I satis es condition (A1) if the Hilbet function of R=a is constant on the open set of ideals a geneated by s foms of the given degees such that codim a : I X s; and (A2) if the Hilbet function of R= a : I is constant on this set. Both conditions ae satis ed automatically fo s W g :ˆ codim I (the case s ˆ g is the theoy of linkage). Thus we will focus on the cases s > g. Ou st esults concen the case s ˆ g 1 and the case of ideals of small codimension (see Theoem 1.1 and Coollay 2.2): THEOREM 0.1. With notation as above, both conditions (A1) and (A2) ae satis ed if. s ˆ g 1 and, fo (A2), I is a complete intesection locally in codimension g; o. codim I ˆ 2 and R=I is Cohen^Macaulay; o. codim I ˆ 3 and R=I is Goenstein. Befoe stating futhe esults we motivate the codimension condition in (A1) and (A2). Recall fom Atin and Nagata [3] that an ideal J in any ing satis es the condition G s if, fo each pime ideal p containing J with codim p W s 1, the minimal numbe of geneatos m J p is at most codim p. If I satis es G s then codim a : I X s when a is geneated by s geneic foms of suf ciently lage degees, and thus the codimension condition is necessay fo a to be in the desied open set. On the othe hand, many inteesting ideals satisfy G s. Fo example any smooth (o even locally complete intesection) pojective vaiety can be de ned (scheme-theoetically) by an ideal satisfying G 1 ^thatis,g s fo evey s. Most of ou esults give cases whee an ideal satisfying G s 1 has popety (A1) and cases whee an ideal satisfying G s has popety (A2). The following gives the avo of what we can pove. A special case of a combination of Theoem 2.1, Poposition 3.1, and Coollay 4.3 (a) says that I satis es (A1) o (A2) if it satis es G s 1 o G s, espectively, and if the depths of the ings R=I j ae not too small fo j W s g 1. In fact we only need to assume the vanishing of a single local cohomology module of each of these ings:

3 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 195 THEOREM 0.2. Let R be a polynomial ing ove a eld with ielevant maximal ideal m, leti R be a homogeneous ideal of codimension g and dimension d, and assume that Hm d j R=I j ˆ0fo 1 W j W s g 1. If I satis es G s 1 o G s, then (A1) o (A2), espectively, hold. Fo example, the vanishing condition is satis ed fo s ˆ g 2 if R=I is Cohen^Macaulay. The distinction between ideals that satisfy the conditions and those that don't is sometimes athe subtle. Fo example the ideal of a Veonese suface P 2,!P 5 (2 by 2 minos of a geneic symmetic 3 by 3 matix) and that of a two-dimensional ational nomal scoll in P 5 (2 by 2 minos of a suf ciently geneal 2 by 4 matix of linea foms) behave diffeently: The ideal of the Veonese satis es (A1) and (A2) fo all s (Theoem 2.1, Poposition 3.1, and [12, 2.3 and 3.3]), wheeas the ideal of the scoll satis es these popeties fo s W 5 ˆ g 2 (Coollay 4.6) but does not satisfy (A1) fo s ˆ 6 (Remak 6.5). To explain these esults, we intoduce ou main de nitions: DEFINITION. Let R be a gaded ing, and let I R be a homogeneous ideal. (a) We say that I is s-pasimonious if the following holds fo each 0 W i W s: fo evey i-geneated homogeneous ideal b I, and evey homogeneous element a 2 I such that we have codim b : I X i and codim b; a : I X i 1 b : I ˆ b : a: (b) We say that I is s-thifty if the following holds fo each 0 W i W s: fo evey i-geneated homogeneous ideal b I, and evey homogeneous element a 2 I such that we have codim b : I X i and codim b : I ; a X i 1 b : I \I ˆ b and a is a nonzeodiviso on R= b : I : The de nition applies to evey ing (fo example to localizations of gaded ings) since we egad othewise ungaded ings as being tivially gaded (evey element has degee 0). The idea behind the names is that if b and a ae suf ciently geneal, so that the codimension conditions hold, then thee ae no `exta' elements that annihilate a (as compaed with I) modulo b. Theoem 2.1 states that if I is G s 1 and s 1 -pasimonious, then (A1) is satis ed, and that if I is G s and s 1 -thifty then (A2) holds. The actual esult also includes

4 196 MARC CHARDIN ET AL. the case whee we know the hypotheses only `locally in codimension W ' and yields only some of the coef cients of the Hilbet polynomial ^ thus it can be used fo pojective vaieties, whee the hypotheses ae ful lled locally on the vaiety. Theoem 2.3 e nes this statement, and computes the changes in Hilbet functions as the degees d i ae changed. Given these esults it is inteesting to have conditions unde which an ideal is s-pasimonious o s-thifty. The est of the pape is devoted to such conditions. We show (Poposition 3.1) that pasimony and thift can be deduced fom the condition that cetain esidual intesections satisfy the popety S 2 of See. This condition of being esidually S 2 has othe applications ^ fo example to d-sequences, and to bounding the codimension of annihilato ideals ^ see Coollay 3.6. Of couse we gain nothing by eplacing the assumptions of pasimony o thift by the condition esidually S 2 unless we can check the latte condition moe easily! In Theoem 4.1, the hadest esult of the pape, we give a suf cient condition fo esidually S 2, essentially in tems of the vanishing of local cohomology ^ fo example in the case of smooth pojective vaieties, this condition becomes the vanishing of cetain cohomology of some powes of the conomal bundle, as in Theoem 0.2 above. In Section 5, we give an inteesting class of esidually S 2 ideals by poving that if X P n k is an isomophic pojection of a educed complete intesection, then the de ning ideal I of X satis es conditions (A1) and (A2) fo s W n. Moe geneally, fo any pojection, we give conditions in tems of the codimension of the conducto ^ see Theoem 5.3. In Section 6 we give a numbe of examples showing that ou cohomological condition fo pasimony is not too fa fom being shap in inteesting cases. A poblem elated to the one above is as follows: Given a homogeneous ideal of a polynomial ing in n vaiables I R, we again choose s elements of I of given degees d i and conside J :ˆ a 1 ;...; a s : I 1 ˆ[ j a 1 ;...; a s : I j : It is easy to see that if the d i ae lage enough then fo geneic choices of the a i the ideal J will have codimension X s, and that fo any a i such that J has codimension X s, theidealj is unmixed of codimension exactly s (o the unit ideal). As befoe, thee is a `geneic' Hilbet function. Now suppose only that the codimension of J is s. Unde what cicumstances can we conclude that the Hilbet function (o Hilbet polynomial o... o the degee) is equal to the geneic one? Fo example, it was discoveed in the 19th centuy by Chasles, Halphen, Schubet (and subsequently poved by Kleiman, [18]) that if I is the ideal of the Veonese suface in the pojective space of plane conics, s ˆ 5, and the a i ae the sextic equations that say that a conic is tangent to 5 given geneal conics, then J is the educed ideal of 3264 points (see [19] fo the histoy of this poblem and fo efeences). Such sextic equations actually span the symbolic squae of I up to an ielevant

5 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 197 component. If we just assume that the a i ae 5 sextics in the symbolic squae geneal enough so that J has codimension 5, is it still tue that the degee of R=J is 3264? Does this condition detemine the Hilbet function of R=J? Answes to these questions might be inteesting geometically. 1. Ideals of Small Codimension In this section we take up the ideals of small codimension mentioned in Theoem 0.1. We do not need the G s conditions that will appea in many othe esults of this pape, and we also pove the stonge esult that the Hilbet functions of the ings R=a and R= a : I ae detemined by that of R=I ^ one does not need to know deepe invaiants. The poof could easily be made into a computation of the new Hilbet functions fom the old one; we hope to etun to such esults in a subsequent pape. THEOREM 1.1. Let R be a polynomial ing ove a eld, let I R be a homogeneous ideal and let a I be a homogeneous s-geneated ideal with codim a : I X s. If I is pefect of codimension 2 o Goenstein of codimension 3, then the Hilbet functions of R=a and R= a : I ae detemined by the Hilbet function of R=I and the degees of the s homogeneous geneatos of a. Poof. We may assume that the gound eld k is in nite and that s X 2. Fist conside the case whee I is pefect of codimension 2. In this case I has a homogeneous minimal esolution of the fom 0! G F 1! f G F 0! I! 0 whee G is the lagest fee summand the two fee modules have in common. Witing the Hilbet seies of R=I in the fom p t = 1 t dim R we see that giving the Hilbet seies is equivalent to giving F 1 and F 0. Let d 1 ;...; d s be the degees of the s geneatos of a, andwiteh ˆ s iˆ1 R d i. We need to show that the Hilbet functions of I=a and of R= a : I ae detemined by the modules F 0 ; F 1 ; H. The R-module I=a has a homogeneous pesentation G F 1 H! c G F 0! I=a! 0; whee c ˆ jj p has size n by n 1 s, say. Now a : I ˆ ann I=a I n c ; hence codim I n c X codim a : I X s ˆ n 1 s n 1 X 2. Thus by [7, 3.1], a : I ˆ I n c. Futhemoe, the Buchsbaum^Rim and Eagon^Nothcott complexes associated to c yield homogeneous fee esolutions of I=a ˆ coke c and of R= a : I ˆR=I n c, espectively. These esolutions show that the Hilbet functions of coke c and of R=I n c ae detemined by the gaded modules F 0 ; F 1 ; H; G; as long as codim I n c X s.

6 198 MARC CHARDIN ET AL. To show the independence fom G, conside the n by n 1 s matix Z ˆ 1G 0 and the family of homogeneous matices c 0 0 l ˆ c lz ˆ l1 G e ; l 2 k. Fol geneal, codim I n c l X s, and we may eplace c by c l without changing F 0 ; F 1 ; H; G. But since l1 G e is invetible fo geneal l, coke c l has a homogeneous pesentation of the fom F 1 H! F 0! coke c l! 0; eliminating the dependence on G. Next, conside the case whee I is Goenstein of codimension 3. By [6, 2.1], I has a homogeneous minimal esolution of the fom 0! R e!g e F e! j G F! I! 0; whee j is an altenating matix, G G e has even ank, and F; e ae detemined by the Hilbet function of R=I. Now one poceeds as above, eplacing [7, 3.1] by [20, 10.5 (a) and its poof], and the complexes of Buchsbaum^Rim and Eagon^Nothcott by the complexes of [20, 10.5 (b),(c)]. 2. Pasimonious Ideals By a `homogeneous ing' (ove k) we shall mean an N-gaded Noetheian ing R whee R ˆ R 0 R 1 Š and R 0 ˆ k is a eld. Wite MŠŠ fo the Hilbet seies of a nitely geneated gaded module M ove a homogeneous ing. We shall say that the Hilbet seies of two nitely geneated gaded modules M; N ove a homogeneous ing R ae -equivalent and wite MŠŠ NŠŠ, if they diffe by the Hilbet seies of R-modules whose annihilatos have codimension at least 1. Note that if dim R ˆ then this means that the Hilbet seies ae equal; while fo smalle values of, it implies that the tems of the Hilbet polynomials of M and N of degees dim R 1;...; dim R 1 coincide. We wite M N if thee is a sequence of homogeneous R-linea maps between M and N that ae isomophisms locally in codimension W. THEOREM 2.1. Let R be a homogeneous ing ove an in nite eld, let I Rbea homogeneous ideal, and let a I be a homogeneous s-geneated ideal with codim a : I X s. (a) If I satis es G s 1 and is s 1 -pasimonious locally in codimension W, then the Hilbet seies of R=a is detemined, up to -equivalence, by I and the degees of the s homogeneous geneatos of a.

7 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 199 (b) If I satis es G s and is s 1 -thifty locally in codimension W, then the Hilbet seies of R= a : I is detemined, up to -equivalence, by I and the degees of the s homogeneous geneatos of a. Theoem 2.1 will be deduced fom Theoem 2.3 below. COROLLARY 2.2. Let R be a homogeneous Cohen^Macaulay ing, let I Rbea homogeneous ideal of codimension g, and let a I be a homogeneous g 1 -geneated ideal with codim a : I X g 1. The Hilbet function of R=a is detemined by I and the degees of the g 1 homogeneous geneatos of a. If moeove IisG g 1 (i.e., a complete intesection locally in codimension g) thenthesameistue fo R= a : I. Poof. Any ideal of codimension g in a Cohen^Macaulay ing is g-pasimonious and g-thifty. Thus afte an extension to make the gound eld in nite we may apply Theoem 2.1 (a) o (b), espectively. Hee is a moe explicit vesion of Theoem 2.1 that allows one to pass fom one a to anothe. This can actually be used to compute Hilbet functions, as will be illustated in the last section. THEOREM 2.3. Let R be a homogeneous ing ove an in nite eld, let I Rbea homogeneous ideal, and let a I be an ideal geneated by s W dim R foms of degees d 1 ;...; d s with codim a : I X s. Let c 1 ;...; c s be foms of degees e 1 ;...; e s contained in I and fo each x ˆfi 1 ;...; i k g sš wite c x ˆ c i1 ;...; c ik. Assume that codim c x : I X jxj fo evey x. (a) (b) If I satis es G s 1 and is s 1 -pasimonious locally in codimension W, then X s R=aŠŠ t d 1 d s e 1 e s kˆ0 1 X Y s k 1 t e j d j R=c x ŠŠ: x sš j62x jxjˆk If I is s 1 -thifty locally in codimension W, and codim I c x : I X jxj 1 fo evey x such that jxj W s 1, then X s R= a : I ŠŠ t d 1 d s e 1 e s kˆ0 1 X Y s k 1 t e j d j R= c x : I ŠŠ: x sš j62x jxjˆk Fo the poof of Theoem 2.3 we will need the following obsevation about thift: LEMMA 2.4. Let R, I, b and a be as in the de nition of thift and let `ö' denote images in R :ˆ R= b : I. IfIiss-thifty,then: (a) b : I ˆ b : a. (b) b; a : I = b : I a : I.

8 200 MARC CHARDIN ET AL. Poof. Fo (a), since the element a is a non zeodiviso modulo b : I, the inclusions a b : a b b : I yield b : a b : I, hence b : a ˆ b : I. Pat (b) holds because b : I ; a : I ˆ b : I \I; a : I ˆ b; a : I: We shall make fequent use of a geneal position lemma: LEMMA 2.5. Let R be a homogeneous ing ove an in nite eld, and let I Rbea homogeneous ideal satisfying G s 1. Let a I be an ideal geneated by foms a 1 ;...; a s of degees d 1 ;...; d s, and assume that codim a : I X s. (a) If d s ˆ minfd i g; then the geneatos a 1 ;...; a s can be chosen so that codim a 1 ;...; a s 1 : I X s 1: (a 0 ) If I satis es G s and d s ˆ minfd i g, then the geneatos a 1 ;...; a s can be chosen so that codim a 1 ;...; a s 1 : I ; a s X s: (b) Fo evey e X maxfd i g; thee exist foms c 1 ;...; c s of degee e in I such that codim c x : I X jxj; fo evey x sš; codim I c x : I X jxj 1; fo evey x sš with jxj W s 2: (b 0 ) If I satis es G s ;then foevey e X maxfd i g;thee exist foms c 1 ;...; c s of degee e in Isuchthat codim c x : I X jxj; fo evey x sš; codim I c x : I X jxj 1; fo evey x sš with jxj W s 1: Now assume that d 1 ˆˆd s ˆ d. Let g I be anothe ideal geneated by foms g 1 ;...; g s of degee d, so that codim g : I X s. (c) The geneatos a 1 ;...; a s and g 1 ;...; g s can be chosen so that fo b i ˆ a 1 ;...; a i 1 ; g i 1 ;...; g s, 1 W i W s; we have codim b i : I X s 1; codim b i ; a i : I X s; codim b i ; g i : I X s: (c 0 ) If I satis es G s ; then the geneatos a 1 ;...; a s and g 1 ;...; g s can be chosen so that fo b i ˆ a 1 ;...; a i 1 ; g i 1 ;...; g s ; 1 W i W s; we have codim b i : I X s 1; codim b i : I ; a i X s; codim b i : I ; g i X s:

9 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 201 Poof. The poof is vey simila to that given in [26, the poof of 1.4]. Poof of Theoem 2.1. By Lemma 2.5 (b) o (b 0 ), thee exist foms c 1 ;...; c s as in Theoem 2.3 (a) o (b), espectively. Now this theoem implies that the Hilbet seies of R=a o R= a : I, espectively, ae detemined, up to -equivalence, by c 1 ;...; c s,by I, and by the degees of the s homogeneous geneatos of a. Poof of Theoem 2.3. We may assume that s X 1. Notice that I satis es G s in pat (b). Choose d X maxfd i ; e i j ig and set t ˆ #fi j d i 6ˆ d o e i 6ˆ dg. Weaegoingto pove both statements by induction on t. If t ˆ 0, then d ˆ d 1 ˆˆd s ˆ e 1 ˆˆe s, and the assetion is that R=aŠŠ R=cŠŠ o R= a : I ŠŠ R= c : I ŠŠ, espectively. We may choose a 1 ;...; a s and c 1 ;...; c s as in Lemma 2.5 (c) o (c 0 ), espectively. (It is possible that we lose some of the hypotheses on the subset ideals c x, but it does not matte since the assetion is simple in this special case.) It suf ces to show that setting b i ˆ a 1 ;...; a i 1 ; c i 1 ;...; c s fo 1 W i W s, we have b i ; a i ŠŠ b i ; c i ŠŠ fo (a) and b i ; a i : IŠŠ b i ; c i : IŠŠ fo (b). But indeed the de nition of pasimony gives b i ; a i =b i R= b i : I d b i ; c i =b i. On the othe hand by Lemma 2.4 (b), thift implies b i ; a i : I = b i : I b i ; c i : I = b i : I, sincea i and c i ae non zeodivisos modulo b i : I and have the same degee. Next assume that t > 0. We may suppose that d s ˆ minfd i g,andthatd s < d o e s < d. Choose a 1 ;...; a s as in Lemma 2.5 (a) o (a 0 ), espectively. Wite a 0 ˆ a 1 ;...; a s 1.Letxbe a linea fom of R which is not in any of the nitely many pimes of codimension s 1 containing a 0 : I, of codimension jxj containing c x : I; jxj W s 1, and, fo (b), of codimension jxj 1 containing I c x : I ; jxj W s 2. Wite y ˆ x d d s and z ˆ x d e s.nowya s and zc s ae foms of degee d. Moeove, codim a 0 ; ya s : I X s o codim a 0 : I ; ya s X s; espectively, and the sequence c 1 ;...; c s 1 ; zc s has the same codimension popeties as c 1 ;...; c s. We st teat pat (a). By the de nition of pasimony we have so that a=a 0 R= a 0 : I d s a 0 ; ya s =a 0 d d s ; R=aŠŠ t d s d R= a 0 ; ya s ŠŠ 1 t d s d R=a 0 ŠŠ: 2:6 Now, fom the induction hypothesis, and setting D i ˆ d 1 d i and E i ˆ e 1 e i,weobtain R= a 0 ; ya s ŠŠ t D s 1 E s 1 X s kˆ0 1 s k X jxjˆk s2x Y 1 t e j d j R= c x fsg ; zc s ŠŠ: j62x

10 202 MARC CHARDIN ET AL. Fomula (2.6) with c x in place of a shows that t es d R= c x fsg ; zc s ŠŠ R=c x ŠŠ 1 t es d R=c x fsg ŠŠ; and substituting into the pevious fomula we get t ds d R= a 0 ; ya s ŠŠ t D s E s X s kˆ1 1 s k X jxjˆk s2x Y 1 t e j d j R=c x ŠŠ 1 t es d R=c x fsg ŠŠ j62x 2 X s ˆ t D s E s 6 kˆ1 1 X Y s k 1 t e j d 4 j R=c x ŠŠ jxjˆk j62x s2x kˆ0 1 X s k 1 t es d Y 1 t e j d j 7 R=c x ŠŠ5: jxjˆk j62x s62x j6ˆs Xs 1 Also by induction hypothesis, X s 1 1 t ds d R=a 0 ŠŠ t D s E s kˆ0 1 X s k t es d t e s d s Y 1 t e j d j R=c x ŠŠ: jxjˆk j62x s62x j6ˆs Taking the sum, using (2.6), and noticing that 1 t e s d t e s d t e s d s ˆ 1 t e s d s, we obtain 2 X s R=aŠŠ t D s E s 6 kˆ1 1 X Y s k 1 t e j d 4 j R=c x ŠŠ jxjˆk j62x s2x Xs 1 kˆ0 1 s k X jxjˆk s62x 3 Y 1 t e j d j 7 R=c x ŠŠ5: This is the fomula asseted in (a). We now tun to the poof of (b). By Lemma 2.4 (b), j62x a : I = a 0 : I a 0 ; ya s : I = a 0 : I d d s ; 3

11 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 203 which yields, R= a : I ŠŠ t d s d R= a 0 ; ya s : I ŠŠ 1 t d s d R= a 0 : I ŠŠ: 2:7 One can now poceed as above, fomally eplacing (2.6) by (2.7). 3. Residually S 2 Ideals Ae Pasimonious and Thifty We know no useful chaacteization of s-pasimonious and s-thifty ideals, but pasimony and thift ae implied by the Atin^Nagata popety studied in [26]. Hee we give bette suf cient conditions. In this section we de ne the notion `s-esidually S 2 ' and exhibit some of its popeties. The advantage of the condition `s-esidually S 2 ' is that it can be checked fom homological popeties of I and its powes. Such a citeion is given in the next section. DEFINITION. Let R be a Noetheian ing, let I R be an ideal of codimension g, let K R be a pope ideal, and let s X g be an intege. (a) (b) K is called an s-esidual intesection of I if thee exists an s-geneated ideal a I such that K ˆ a : I and codim K X s. K is called a geometic s-esidual intesection of I if K is an s-esidual intesection of I and if in addition codim I K X s 1. We shall often wite `K ˆ a : I is an s-esidual intesection of I' to indicate that the conditions of the above de nition ae satis ed. DEFINITION. Let R be a Noetheian ing, let I R be an ideal of codimension g, and let s be an intege. (a) (b) I is said to be s-esidually S 2 if fo evey i with g W i W s and evey i-esidual intesection K of I, R=K is S 2. I is said to be weakly s-esidually S 2 if fo evey i with g W i W s and evey geometic i-esidual intesection K of I, R=K is S 2. These popeties ae weake vesions of the popeties AN s and ANs of [26], whee the esidual intesections ae equied to be Cohen^Macaulay instead of meely S 2. Notice that if s W g 1thenI is automatically s-esidually S 2. We can now state the main esult of this section: PROPOSITION 3.1. Let R be a gaded Cohen^Macaulay ing, and let I Rbea homogeneous ideal. If I satis es G s and is weakly s 1 -esidually S 2,thenIis s-pasimonious and s-thifty. Poof. The statement follows fom Coollay 3.6 (a).

12 204 MARC CHARDIN ET AL. SOURCES OF EXAMPLES. The hypotheses of Poposition 3.1 ae ful lled if I satis es G s and if moeove, afte localizing, I has the sliding depth popety o, moe estictively, is stongly Cohen^Macaulay (which means that fo evey i; the i-th Koszul homology H i of a geneating set h 1 ;...; h n of I satis es depth H i X dim R n i o is Cohen^Macaulay, espectively) ([12, 3.3], [17, 3.1]). The latte condition always holds if I is a Cohen^Macaulay almost complete intesection o a Cohen^Macaulay deviation 2 ideal of a Goenstein ing ([4, p.259]). It is also satis ed fo any ideal in the linkage class of a complete intesection ([16, 1.11]). Standad examples include pefect ideals of codimension 2 ([2], [10]) and pefect Goenstein ideals of codimension 3 ([27]). We shall often use the following emak on geneal position: LEMMA 3.2 (e.g. [3, 2.3] o [26, 1.6 (a)]). Let R be a Noetheian local ing, let a I R be ideals, and assume that a is s-geneated with codim a : I X s and that Isatis esg s. Then thee exists a geneating sequence a 1 ;...; a s of a such that with a i ˆ a 1 ;...; a i and K i ˆ a i : I; codim K i X iandcodim I K i X i 1 wheneve 0 W i W s 1. The next esult contains basic facts about pasimony: PROPOSITION 3.3. Let R be a Noetheian gaded ing, and let I Rbeahomogeneous ideal. (a) (b) I is s-pasimonious if and only if fo evey i; b; as in the de nition of pasimony, b : I is unmixed of height i (o b : I ˆ R). If I is s-pasimonious then fo evey a, b as in the de nition of thift, a is a non zeodiviso on R= b : I. Poof. It suf ces to pove that if I is s-pasimonious and b : I 6ˆ R, thenb : I is unmixed of height i. The convese is obvious and pat (b) follows immediately fom (a). Thus, let p R be a pime of height X i 1 that contains b : I. Choose an element x 2 p not in any minimal pime of b : I of height i. Now codim b; ax : I X i 1, hence by the pasimony of I, b : I ˆ b : ax ˆ b : a : x b : I : x: Hence x 2 p is a nonzeodiviso on R= b : I, showing that p cannot be an associated pime of b : I: The next esult shows that the unmixedness condition of the pevious poposition is always satis ed by weakly s 1 -esidually S 2 ideals. This genealizes [26, 1.7] (which, in tun, extends pats of [17, 3.1]).

13 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 205 PROPOSITION 3.4. Let R be a Cohen^Macaulay ing, let I R be an ideal, and assume that I is G s and weakly s 1 -esidually S 2. Fo evey s-esidual intesection K ˆ a : I of I one has: (a) K is unmixed of codimension s. (b) The associated pimes of a have codimension at most s. (c) If K is a geometic esidual intesection, then K \ I ˆ a. Poof. Afte localizing the poof is the same as that of [26, 1.7]. COROLLARY 3.5. Let R be a Cohen^Macaulay ing, let I R be an ideal satisfying G s,andleti 0 I be an ideal that agees with I locally up to codimension s. If I is s-esidually S 2 (espectively weakly s-esidually S 2 )thensoisi 0. Poof. Poposition 3.4 (a) implies that fo codim I W i W s, evey i-esidual intesection of I 0 is also an i-esidual intesection of I, and one is geometic iff the othe is. COROLLARY 3.6. Let R be a Cohen^Macaulay ing, and let I R be an ideal satisfying G s. (a) If I is weakly s 1 -esidually S 2 ; then I is s-pasimonious and s-thifty. (b) Suppose that R is local and I is weakly s 2 -esidually S 2 : If a I is an s-geneated ideal with codim a : I X s, and a 1 ;...; a s satisfy the conditions of Lemma 3.2, then a 1 ;...; a s is a d-sequence elative to I (that is, a 1 ;...; a i : a i 1 \I ˆ a 1 ;...; a i fo 0 W i W s 1 : (c) Suppose that I is weakly s 1 -esidually S 2 : If a = I is an ideal geneated by s elements, then codim a : I W s. Poof. (a) The ideal I is s-pasimonious by Popositions 3.3 (a) and 3.4 (a), and then I is s-thifty by Popositions 3.3 (b) and 3.4 (c). (b) Let 0 W i W s 1andwitea i ˆ a 1 ;...; a i. Notice that codim a i : I ; a i 1 X i 1. By pat (a) of this coollay, I is s 1 -pasimonious and s 1 -thifty. Theefoe a i : a i 1 \I ˆ a i : I \I ˆ a i. (c) We may suppose that codim a : I X s, and then use Poposition 3.4 (a). 4. A Su cient Condition Fo Residually S 2 We now tun to the main technical esult of this pape: A suf cient condition fo an ideal I of codimension g to be s-esidually S 2. This condition is moe geneal than the one of [26, 2.9] (which is a suf cient condition fo the stonge Atin^Nagata popety) as it only equies the vanishing of s g 1 local cohomology modules. The condition involves the vanishing of cetain Ext i R R=I n ; R. Occasionally this vanishing holds not fo I n but fo an ideal equal to I n up to a cetain codimension, and this is sometimes enough. To fomalize this possibility, we make a de nition:

14 206 MARC CHARDIN ET AL. DEFINITION. Let R be a Noetheian ing, and let I R be an ideal of codimension g.wewillsaythatidealsi 1 ;...; I ae good appoximations of the st powes of I if the following two conditions ae satis ed: (a) I j coincides with II j 1 locally up to codimension g j 1wheneve1WjW. (b) I j coincides with I j locally up to codimension g j 1 wheneve 2 W j W 1. Fo convenience we set I j ˆ I j ˆ R if j W 0. Note that I j need not contain I j 1 (but in pactice they often do). Fo example, one may choose I j to be (a) I j ;o (b) I j W g 1 (hee J W i denotes the intesection of all pimay components of codimension at most i, foj R any ideal); o (c) I j W minfg j;g 1g ;o (d) I j W g j 1 in case I j has no associated pimes of codimension g j fo 1 W j W 1; o (e) I j in case I j has no embedded associated pimes of codimension at most minfg j; g 1g fo 1 W j W (hee I j denotes the j-th symbolic powe). THEOREM 4.1. Let R be a Goenstein ing, and let I R be an ideal of codimension g satisfying G s fo some s X g. Suppose that I 1 ;...; I s g 1 ae good appoximations of the st s g 1 powes of I locally at evey maximal ideal containing I. If Ext g j R R=I j; R ˆ0 fo 1 W j W s g 1, then I is s-esidually S 2. COROLLARY 4.2. Let R be a local Goenstein ing, and let I Rbeanidealof codimension g satisfying G s fo some s X g. Let I 1 ;...; I s g 1 be ideals such that I j coincides with I j locallyuptocodimensionminfg j; sg. IfExt g j R R=I j; R ˆ0 fo 1 W j W s g 1, then I is s-esidually S 2. Fo applications to pojective vaieties it is convenient to efomulate Theoem 4.1 in tems of local cohomology: COROLLARY 4.3. Let R; m be a local Goenstein ing, and let I R be an ideal of codimension g satisfying G s fo some sx g. Set d ˆ dim R=I. Suppose that I 1 ;...; I s g 1 ae good appoximations of the st s g 1 powes of I. The ideal Iiss-esidually S 2 if eithe (a) (b) R=I j ˆ0 fo 1 W j W s g 1; o we have containments I 1 I s g 1 and Hm d i g 1. H d j m I j 1=I j ˆ0 fo 1 W j W i W s The conditions of Coollay 4.3 ae satis ed in paticula if depth R=I j X d j 1 fo 1 W j W s g 1,whichintunholdsifI is stongly Cohen^Macaulay (assuming that I satis es G s ) ([11, the poof of 5.1]).

15 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 207 COROLLARY 4.4. (cf. also [23,4.1]). Let R; m be a local Goenstein ing, let I R be an ideal of codimension g, and let I 1 ˆ I W g.ifhm d 1 R=I 1 ˆ0whee d ˆ dim R=I, then I is g-esidually S 2. COROLLARY 4.5. Let R; m be a egula local ing which is essentially of nite type ove a pefect eld k, and let I R be an ideal of codimension g. Wite I 1 ˆ I W g 1 ; A ˆ R=I 1,anddˆdim A. Assume that A is educed and a complete intesection locally in codimension 1 (in A). If Hm d 1 A ˆHd 2 m A ˆ0 and Hm d 3 O k A ˆ 0, theniis g 1 -esidually S 2. Poof. The standad exact sequence 0! I 1 =I 2 1! O k R R A A! O k A!0 shows that we may apply Coollay 4.3 (b) taking I 1 ; I 2 1 as I 1 ; I 2. Hee ae the consequences fo Hilbet functions: COROLLARY 4.6. Let R be a homogeneous Goenstein ing, let I Rbeahomogeneous ideal of codimension g satisfying G g 1,andletaI be a homogeneous g 2 -geneated ideal with codim a : I X g 2. Let I 1 ˆ I W g,letmbe the ielevant maximal ideal of R, and wite d ˆ dim R=I.IfHm d 1 R=I 1 ˆ0, then the Hilbet function of R=a is detemined by I and the degees of the g 2 homogeneous geneatos of a. If moeove I satis es G g 2, the same is tue fo the Hilbet function of R= a : I. Poof. The assetion follows fom Coollay 4.4, Poposition 3.1, and Theoem 2.1. COROLLARY 4.7. Let R be a polynomial ing ove a pefect eld k, let I Rbea homogeneous ideal of codimension g, and let a I be a homogeneous g 3 -geneated ideal with codim a : I X g 3. Let I 1 ˆ I W g 1, let m be the ielevant maximal ideal of R, and wite A ˆ R=I 1 and d ˆ dim A. Assume that A is educed and a complete intesection locally in codimension 1. If Hm d 1 A ˆHd 2 m A ˆ0 and Hd 3 m O k A ˆ 0, then the Hilbet function of R=a is detemined by I and the degees of the g 3 homogeneous geneatos of a. If moeove Isatis esg g 3, the same is tue fo the Hilbet function of R= a : I : Poof. One uses Coollay 4.5, Poposition 3.1, and Theoem 2.1. We next tun to some lemmas necessay fo the poof of Theoem 4.1. The st is an easy consequence of the change^of^ings spectal sequence (o one can simply ague using injective esolutions): LEMMA 4.8. Let R! S be a homomophism of ings, and let M be an R-module with Ext j R S; M ˆ0 fo j < g. Then fo evey intege n and evey S-module N, thee is a

16 208 MARC CHARDIN ET AL. natual homomophism Ext n S N; Extg R S; M! Extn g R N; M ; which is an isomophism if n ˆ 0 o if Ext j R S; M ˆ0 fo j 6ˆ g. The following povides a cucial step in the poof of Theoem 4.1: LEMMA 4.9. (cf. also [26, 2.1]). Let R be a Noetheian local ing satisfying S 2, assume that R has a canonical module o ˆ o R, and wite _ ˆ Hom ; o. Let I R be an ideal, let a be an R-egula element contained in I, let K ˆ a : I 6ˆ R, and wite R ˆ R=K. We have Ext 1 R R; o Io =ao: Poof. Thee ae natual isomophisms a : I Hom I; a Hom I; R a Hom I; Hom o; o a Hom I R o; o a Hom Io; o a; which yield K _ ˆ a 1 Io. Now applying _ to the exact sequence 0! K! R! R! 0 we get an exact sequence 0! o! a 1 Io! Ext 1 R R; o!0: Thus Ext 1 R R; o a 1 Io =o, and the desied esult follows upon multiplication by a. PoofofTheoem4.1.Let K ˆ a : I be any s-esidual intesection of I. Wemay assume that R is local. Let K i be ideals as in Lemma 3.2 and wite R i ˆ R=K i. The theoem is a consequence of the following assetions, which we shall pove by induction on i: (a) R i satis es S 2 fo 0 W i W s; (b) ExtR i 1 i g 2R i ; R ˆ0 fo 0 W i W s 1; (c) o Ri I i g 1 R i fo 0 W i W s 1, whee _ ˆ Hom ; o Ri ; note that this notation implicitly uses the value of i. We st show that if s > 0 (as will be the case in the poof of pats (b) and (c)) then I j R 0 I j fo all j X 1. Equivalently, I j \ 0 : I ˆ0foallj X 1. Indeed, since R is

17 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 209 unmixed it suf ces to pove this afte localizing at a pime p of R such that dim R p ˆ 0. Since s > 0, the ideal I satis es G 1,soI p is 0 o the unit ideal. By the de nition of good appoximations, we have I j p I p, and the fomula follows. To pove (a), (b), (c), st let i ˆ 0. In this case ou assetion is only nontivial if g ˆ 0. So let g ˆ 0 and wite ˆ Hom ; R. As fo (a), notice that 0 : I ˆ 0 : I 1 R=I 1.SinceExt 1 R R=I 1; R ˆ0, applying to a fee R-esolution!F 2! F 1! F 0 ˆ R of R=I 1, yields an exact sequence 0! 0 : I! R! F 1! F 2 ; which shows that R 0 ˆ R= 0 : I satis es S 2. As fo (b) and (c), notice that s > 0, so I 2 R 0 I 2, which gives (b). Futhemoe I 1 R 0 I 1, thus I 1 R 0 I 1 R 0 I1.AsI 1 is unmixed locally in codimension one, I 1 and 0 : 0 : I 1 coincide locally in codimension one. Hence I1 0 : 0 : I 1.But 0: 0: I 1 0 : 0 : I 1 ˆ0: 0: I, and the latte module is o R0. We now pefom the induction step fom i X 0toi 1.. Fo (a) we may suppose i W s 1. By the induction hypothesis, I is i-esidually S 2. Popositions 3.4 (a) and 3.1 and Lemma 2.4 (b) imply that R i satis es S 2 and is equidimensional of codimension i in R, a i 1 is egula on R i, K i 1 R i ˆ a i 1 R i : IR i Hom IR i ; R i, and depth I o Ri > 0. Since II i g 1 and I i g 2 coincide locally in codimension i 1, pat (c) fo i shows that Io Ri _ I i g 2 R i _. Putting this togethe, one obtains natual isomophisms, K i 1 R i Hom IR i ; R i Hom IR i Ri o Ri ; o Ri (as R i is S 2 Hom Io Ri ; o Ri (as depth I o Ri > 0 Hom I i g 2 R i ; o Ri (by the emak above) Ext i R I i g 2R i ; R (by Lemma 4.8). By pat (b) fo i, Ext i 1 R I i g 2R i ; R ˆ0. On the othe hand since R i has codimension i, Ext`R I i g 2R i ; R ˆ0 wheneve ` W i 1. Thus, dualizing a fee R-esolution of I i g 2 R i into R, one sees that the R i -module Ext i R I i g 2R i ; R satis es S 3. Theefoe K i 1 R i is S 3, and hence R i 1 R i =K i 1 R i satis es S 2. This concludes the poof of (a) fo i 1.. Fo (b) and (c) we may suppose i 1 W s 1. Recall that by Coollay 3.6 (b) and Poposition 3.1, a 1 ;...; a i 1 fom a d-sequence and a k is egula on R k 1 wheneve 1 W k W i 1. Thus fo 1 W k W i 1andj W s g 1thee ae complexes C kj : 0! I j 1 R k 1! a k I j R k 1 a k I j 1 R k 1! I j R k! 0

18 210 MARC CHARDIN ET AL. having nontivial homology at most in the middle. Call this homology H and notice that H ˆ K k \ I j a k I j 1 = K k 1 \ I j a k I j 1 a k I j 1 : We st claim that Ext`R H; R ˆ0 wheneve ` W minfg j 1; i 1g 4:10 o equivalently H p ˆ 0 wheneve dim R p W minfg j 1; i 1g: To see this let p 2 Spec R with dim R p W minfg j 1; i 1g. Noticethat II j 1 p ˆ I j p ˆ I j p. If I 6 p, then I j 1 6 p; I j 6 p, and K k p ˆ a 1 ;...; a k p. Hence H p ˆ 0 in this case. Next assume that I p. Then j X 1 since g W dim R p W g j 1, and I p ˆ a 1 ;...; a i 1 p since dim R p W i 1 W s 1. Now K k \ I j a k I j 1 p ˆ K k \ I j p. Since j X 1 and a 1 ;...; a k ;...; a i 1 fom a d-sequence geneating I p, we conclude that K k \ I j p ˆ a 1 ;...; a k \I j p ˆ a 1 ;...; a k I j 1 p (cf. [15, Theoem 2.1]) K k 1 \ I j a k I j 1 p : ˆ K k 1 \ I j a k I j 1 p : The vanishing of H p will follow once we have shown that I j 1 p I j 1 p,fo which we may assume j X 2. By assumption I j 1 and I j 1 coincide locally in codimension g j 2. Since Ext g j 1 R R=I j 1 ; R ˆ0, the ideal I j 1 has no associated pimes of codimension g j 1. Theefoe I j 1 p I j 1 p.this concludes the poof of 4:10.. We now tun to the poof of (b) fo i 1 W s 1. We show, by induction on k, 0 W k W i 1, that Ext g j 1 R I j R k ; R ˆ0 wheneve k g 2 W j W i g 3. Statement (b) follows if we set k ˆ i 1. Fist suppose k ˆ 0. ^IfjW0, then g X 2, hence R 0 ˆ R and I j R 0 ˆ RR 0 ˆ R. ^IfjX1, then I j R 0 I j by the emak at the beginning of the poof. Now the assetion follows because Ext g j 1 R I j ; R ˆ0fo g 2 W j W s g 1. Next suppose 1 W k W i 1. Assuming k g 2 W j W i g 3, as in the desied fomula, we have ^Ext g j 2 R H; R ˆ0by 4:10, ^Ext g j 2 R I j 1 R k 1 ; R ˆExt g j 1 R I j R k 1 ; R ˆ0 by induction hypothesis, ^Ext g j 1 R I j R k 1 a k I j 1 R k 1 ; R embeds into Ext g j 1 R I j R k 1 ; R,sincea k 2 I and locally in codimension g j 1, I j and II j 1 coincide. Now using the complex C kj we deive Ext g j 1 R I j R k ; R ˆ0, poving (b).

19 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 211. Finally, we pove (c) fo i 1 W s 1. By pat (a) and Poposition 3.4 (a), we know that R i and R i 1 satisfy S 2 and ae equidimensional of codimension i and i 1 in R. Also, by Poposition 3.1 and Lemma 2.4 (b), K i 1 R i ˆ a i 1 R i : IR i and a i 1 is egula on R i. Wite ˆ Hom ; o Ri and _ ˆ Hom ; o Ri 1. Conside the complex C i 1;i g 2 : 0! I i g 1 R i! a i 1 I i g 2 R i a i 1 I i g 1 R i! I i g 2 R i 1! 0; intoduced above, which has nontivial homology H at most in the middle. By 4:10, Ext`R H; R ˆ0fo`Wi 1, and by pat (b), Exti 1 R I i g 2R i ; R ˆ0. Notice that, since a i 1 2 I and I i g 2 coincides with II i g 1 locally in codimension i 1, the map Ext`R I i g 2R i a i 1 I i g 1 R i ; R!Ext`R I i g 2R i ; R is a monomophism fo ` W i 1 and is an isomophism fo ` W i. Futhe, Ext i R I i g 2R i 1 ; R ˆ0. Thus the above complex induces an exact sequence 0! Ext i R I i g 2R i ; R! Ext i R I i g 1R i ; R! Ext i 1 R I i g 2R i 1 ; R! 0; ko ko ko I i g 2 R i I i g 1 R i I i g 2 R i 1 _ whee the vaious identi cations ae special cases of Lemma 4.8. Dualizing again, we obtain an exact sequence 0! I i g 1 R i! a i 1 I i g 2 R i! I i g 2 R i 1! Ext i 1 R I i g 1R i ; R : This shows that I i g 2 R i =a i 1 I i g 1 R i is isomophic to an R i 1 -submodule of I i g 2 R i 1, and that both modules coincide locally in codimension one in R i 1. Thus, since R i 1 is S 2, I i g 2 R i 1 I i g 2 R i =a i 1 I i g 1 R i, and we must show that the latte module is isomophic to o Ri 1. Now I i g 2 and II i g 1 coincide locally in codimension i 1andR i satis es S 2, hence by ou induction hypothesis, I i g 2 R i =a i 1 I i g 1 R i Io Ri = a i 1 o Ri. But Lemma 4.9 shows the latte module is isomophic to Ext 1 R i R i 1 ; o Ri. Thus it emains to pove that Ext 1 R i R i 1 ; o Ri o Ri 1. Now Lemma 4.8 yields a natual map Ext 1 R i R i 1 ; o Ri!o Ri 1, which is an isomophism locally in codimension one in R i 1 because R i satis es S 2. Thus, since R i 1 is S 2, we get Ext 1 R i R i 1 ; o Ri o Ri 1 o Ri 1.

20 212 MARC CHARDIN ET AL. 5. A Class of Residually S 2 Ideals In this section we illustate how geneal pojections can poduce an abundance of ideals that ae s-esidually S 2, but in geneal fail to satisfy the stonge conditions AN s o AN s. THEOREM 5.1. Let k be an algebaically closed eld, and let Y P n t k be a educed complete intesection of dimension d. Conside a geneal pojection X P n k of Y, and let I k X 0 ;...; X n Š be the (satuated) ideal de ning X. Let d e be the dimension of the closed subset fy 2 Y j edim O Y;y X eg, and wite s ˆ 2n 3 maxfn 1; 2d; d e e 1g (using the convention that dim 1 ˆ 1). Then I is e s-esidually S 2. In paticula, if Y is nonsingula (o, moe geneally, if edim B q W 2 dim B q 1 fo evey nonmaximal homogeneous pime q of the homogeneous coodinate ing B of Y), then I is minfn 2; 2n 2d 3g-esidually S 2. Poof. Let A; B be the homogeneous coodinate ings of X; Y espectively. The geometic condition implies that the conducto of A B has codimension at least minfd 1; n d; n d d e e 1g: The theoem thus follows fom the moe geneal esult in Theoem 5.3. e Remak 5.2. If in the setting of Theoem 5.1, Y P n t k is nondegeneate and nonsingula with t X 1; d X 1andn dx2, then I does not satisfy ANn d. Poof. Let A; B be the homogeneous coodinate ings of X; Y. Since the conducto of the extension A B has codimension X min fd 1; n dg X 2andA6ˆ B, it follows that A ˆ k X 0 ;...; X n Š=I cannot be Cohen^Macaulay. As I is an unmixed adical ideal of codimension n d, thee has to exist a geometic link of I that is not Cohen^Macaulay. To state ou moe geneal esult we eplace the complete intesection above by an ideal J whose conomal module has symmetic powes with suf ciently high depths. This condition is automatically satis ed if J is an unmixed locally stongly Cohen^Macaulay ideal satisfying cetain conditions on the local numbes of geneatos ([11, the poof of 5.1]). THEOREM 5.3. Let k be a pefect eld, let R S be egula domains that ae k-algebas essentially of nite type, and set t ˆ tdeg R S. Let J be an ideal of codimension g in S satisfying G s t 1.SetIˆ J \ R, conside A ˆ R=I B ˆ S=J, and let C ˆ A : A B denote the conducto. Assume that B is educed and a complete intesection locally in codimension 1 (in B) and that codim A C X s t g 3. If pojdim S Sym B j J=J2 W g jfo0 W j W s t g, then I is s-esidually S 2. Poof. We induct on s, the assetion being tivial fo s W 1. Notice that I is the intesection of the contactions of all minimal pimes of J. Now, eplacing R; S; A; B by af ne domains and computing dimensions, one easily

21 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 213 sees that codim I X g t. Thus nothing is to be shown if s < g t,andwemayfom now on assume that s X g t. But then by ou assumption, codim A C > 0 and hence C contains a non zeodiviso of A, which implies that B is a nite A-module. Since s X g t, B is a pefect S-module of gade g by ou assumption, and theefoe evey minimal pime of J has the same codimension g. LocalizingR we may suppose that R; m is a local ing. Moeove, if p 2 V I then thee exists q 2 V J with q \ R ˆ p, and fo evey such q the dimension fomula yields dim S q ˆ dim R p t since the esidue eld extension of R p! S q is algebaic. Applying this equality to any minimal pime p of I one sees that dim R p ˆ dim S q t ˆ dim S q dim B q t ˆ g t. Thus evey minimal pime of I has the same codimension g t. On the othe hand, if we choose q to be the peimage of any maximal ideal n of B, then dim B n ˆ dim S q dim J q ˆ dim R t g ˆ dim R codim I ˆ dim A, o equivalently, the codimension of evey maximal ideal of B is dim A. Finally, the peimage of C in R has codimension at least s t g 3 codim I ˆ s 3, showing that the R-module B=A vanishes locally in codimension W s 2. Let M and N be nitely geneated R-modules. We wite M N if thee is an R-linea map between M and N that is an isomophism locally in codimension W in R. Notice that if M N then H i m M Hi m N as long as i X dim R 1. Afte these pepaatoy emaks we ae now going to pove that I satis es G s 1.To this end let p 2 V I with dim R p W s. Choose q 2 V J with q \ R ˆ p. Since dim S q ˆ dim R p t W s t, we have m J q W dim S q, o equivalently, the deviation d B q of B q is at most dim B q. Now the equality A p ˆ B q yields m I p ˆ codim I p d A p ˆ g t d B q W g t dim B q ˆ codim J q dim B q t ˆ dim S q t ˆ dim R p : Thus I satis es G s 1. Wite d ˆ dim A. Having established the popety G s, we know fom Coollay 4.3 (b) that I is s-esidually S 2 once we show H d i m I j 1 =I j ˆ0 fo 1 W j W i W s t g 1: 5:4 So let 1 W j W s t g 1 and set M ˆ Sym B j 1 J=J2. Fo evey maximal ideal n of B, ou assumption on J implies depth Bn M n X dim B n j 1 ˆ d j 1. Thus the Rad B -depth of the B-module M is at least d j 1, which yields H d i Rad B Since M ˆ0aslongasi X j. This shows that H d i m SymB j 1 J=J2 ˆ 0 fo 1 W j W i W s t g 1: 5:5 A B and d 1 ˆ dim R g t 1 X dim R s 2 1, we have s 2 B, and hence (5.4) follows fom (5.5) if s ˆ g t. Thus we Hm d 1 A Hd 1 m may fom now on assume that s X g t 1. But then by ou assumption, pojdim S J=J 2 W g 1. Since J is a complete intesection locally in codimension g 1 in S, we conclude that the B-module

22 214 MARC CHARDIN ET AL. J=J 2 is tosionfee. The Zaiski Sequence associated to the homomophisms k! S! B yields an exact sequence of B-modules T 1 S=k; B ˆ0! T 1 B=k; B!T 1 B=S; B ˆJ=J 2! T 0 S=k; B ˆO k S S B! T 0 B=k; B ˆO k B!0; whee T 0 S=k; B is pojective and hence fee ([21, and 3.1.5]). Since B is educed, the B-module T 1 B=k; B is tosion ([21, and 3.1.5]), and hence T 1 B=k; B ˆ0 by the tosionfeeness of J=J 2.InpaticulaJ=J 2 is a st syzygy module in a fee esolution of T 0 B=k; B. Likewise, the mophisms k! R! A give ise to two shot exact sequences of B-modules 0! T 1 A=k; B! T 1 A=R; B ˆI=I 2 A B! U! 0 0! U! T 0 R=k; B!T 0 A=k; B! 0; whee T 0 R=k; B is fee. Notice that U is a st syzygy module in a fee esolution of T 0 A=k; B. Finally, fom the homomophisms k! A! B we obtain the Zaiski Sequence T i 1 B=A; B!T i A=k; B!T i B=k; B!T i B=A; B : If p 2 Spec R with dim R p W s 2, then T i B=A; B p T i B p =A p ; B p ˆ T i A p =A p ; B p ˆ0 fo evey i ([21, and 3.1.1]). Thus T i B=A; B 0; which s 2 implies T i A=k; B T i B=k; B : s 2 We st make use of the identi cation T 0 A=k; B T 0 B=k; B : Fom it we s 2 obtain two shot exact sequences 0! K! T 0 A=k; B!V! 0 0! V! T 0 B=k; B!L! 0; whee K 0 L. Compaing st syzygy modules in fee B-esolutions, we conclude that U and syz 1 K syz 1 V ae stably isomophic, and likewise fo J=J 2 s 2 s 2 and syz 1 V syz 1 L. ThusU syz 1 L and J=J 2 syz 1 K ae stably isomophic, which allows us to assume that U is a diect summand of J=J 2 syz 1 K. But syz 1 K F fo some fee B-module F, and hence s 2 Sym B j 1 U,! Sym B j 1 J=J2 syz 1 K Sym B j 1 J=J2 F s 2 j nˆ1 SymB n 1 J=J2 B Sym B j n F : Now (5.5) implies Hm d i SymB j 1 U ˆ 0 fo 1 W j W i W s t g 1: 5:6

23 HILBERT FUNCTIONS, RESIDUAL INTERSECTIONS, AND RESIDUALLY S 2 IDEALS 215 Next we make use of the fact that T 1 A=k; B T 1 B=k; B ˆ0. Fom it we conclude I=I 2 A B U,andthus s 2 s 2 Sym A j 1 I=I 2 Sym B j 1 I=I 2 A B Sym B j 1 U : s 2 s 2 Hence by (5.6), H d i m SymA j 1 I=I 2 ˆ 0 fo 1 W j W i W s t g 1: 5:7 We saw that I satis es G s 1, and by ou induction hypothesis, I is s 1 -esidually S 2. Now Coollay 3.6 (b) shows that locally in codimension s, I can be geneated by a d-sequence. But then in the natual exact sequence 0! N! Sym A j 1 I=I 2! I j 1 =I j! 0; we have N s 0 ([14, Theoem 3.1]). Now (5.7) implies (5.4). 6. Examples Fist we wish to illustate how the fomulas of Theoem 2.3 can be used fo explicit computations of Hilbet functions. EXAMPLE 6.1. Let k be a eld, let R ˆ k X 1 ;...; X 6 Š,andletI R be the de ning ideal of the Veonese suface in P 5 k, that is, the ideal geneated by the 2 by 2 minos of the geneic symmetic matix 0 X 1 X 2 1 X X 2 X 4 X 5 A: X 3 X 5 X 6 We wish to compute R= a : I ŠŠ, wheea I is an abitay ideal geneated by 5 foms of degee d ˆ e 2sothatht a : I X 5. Fo this we may assume that k is in nite. Let c 1 ;...; c 6 be quadics contained in I that satisfy the conditions of Lemma 2.5 (b'), and wite c i ˆ c 1 ;...; c i fo 0 W i W 5. Recall that I is (weakly) 6-esidually S 2 by [12, 2.3 and 3.3]. Thus Poposition 3.1 and Theoem 2.1 (b) imply that, fo evey 0 W i W 5 and evey x 5Š with jxj ˆi, R= c x : I ŠŠ ˆ R= c i : I ŠŠ. Futhemoe, again by Poposition 3.1, c i : I ˆ c i : c i 1, which yields R= c i : I ŠŠ ˆ R= c i : c i 1 ŠŠ ˆ c i 1 =c i ŠŠt 2. Finally, by Poposition 3.4 (a) (o [12, 3.3]), ht c i : I ˆi. Nowfo0WiW2, c i : I ˆ c i is a complete intesection, wheeas by linkage theoy, R= c 3 : I ŠŠ ˆ 1 3t.SinceI has a linea pesentation matix, 1 t 3 c 5 : I is a complete intesection of 5 linea foms, hence R= c 5 : I ŠŠ ˆ 1 1 t. Finally, R= c 4 : I ŠŠ ˆ c 5 =c 4 ŠŠt 2 ˆ R=c 3 ŠŠ c 4 =c 3 ŠŠ I=c 5 ŠŠ R=IŠŠ t 2 ˆ R=c 3 ŠŠt 2 R= c 3 : I ŠŠ R= c 5 : I ŠŠ R=IŠŠt 2 ˆ 1 t : 1 t 2

24 216 MARC CHARDIN ET AL. Now using this infomation in the fomula of Theoem 2.3 (b) yields, R= a : I ŠŠ ˆ X5 ˆ kˆ0 5 t ke 1 t e 5 k R= c k : I ŠŠ k t5e 1 t 5 t4e 1 t e 1 t 1 t 2 10 t3e 1 t e 2 1 3t 1 t 3 10 t2e 1 t e 3 1 t 2 1 t 4 5 te 1 t e 4 1 t 1 t 5 1 te 5 1 t 6 ˆ 1 5t 15t2 6t 5 d 2 1 t fo d X 3 : In paticula one can see that the egulaity of R= a : I is 5 d 2 and the initial degee of a : I is d fo d X 3. Also the degee of R= a : I is 1 10e 40e 2 40e 3 10e 4 e 5 ˆ d 5 40d 2 90d 51. Using the coesponding fomula fo R=aŠŠ one can also check that the minimal degee of an element in a : I that is not in a is 3d 4fod X 3(and1fod ˆ 2). Ou main theoems say that unde vaious hypotheses the Hilbet functions of R=a o R= a : I ae detemined by I and the degees of the geneatos of a.inafewcases (Theoem 1.1) we have seen that a knowledge of the Hilbet function of I alone suf ces. Hee is an example that shows this is not tue in geneal: EXAMPLE 6.2. Let k be a eld, let R ˆ k X 1 ;...; X 6 Š, and again let I R be the de ning ideal of the Veonese suface in P 5 k, that is, the ideal geneated by the 2 by 2 minos of the geneic symmetic matix. Let I 0 be the de ning ideal of the geneic ational nomal scoll of degee 4 in P 5 k, that is, the ideal of 2 by 2 minos of the matix X 1 X 2 X 4 X 5 : X 2 X 3 X 5 X 6 Notice that R=I and R=I 0 have the same Hilbet function. Let a I and a 0 I 0 be ideals geneated by 4 quadics so that codim a : I X 4andcodim a 0 : I 0 X 4. By Coollay 2.2 the Hilbet function of R= a 0 : I 0 is detemined by I 0.Thuswe may pefom a diect computation on a paticula choice of a 0 to obtain R= a 0 : I 0 1 2t t2 ŠŠ ˆ 1 t 2 : On the othe hand, by Example 6.1 we have R= a : I ŠŠ ˆ 1 t = 1 t 2. Next we illustate the fact that ou esults about Hilbet functions do not hold in geneal without pasimony condition o some esidual S 2 assumption.