Purity of branch and critical locus Rolf Ka¨llstro¨m July 6, 2010 0 1 0 2 l Abstract u J To a dominant morphism X/S → Y/S of Nœtherian integral S-schemes one has the in- 5 clusion CX/Y ⊂ BX/Y of the critical locus in the branch locus of X/Y. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of rela- ] tivedifferentials Ω ,Ω ,andΩ thatimply boundson thecodimensions of C and G X/Y X/S Y/S X/Y B . These bounds generalise to a wider class of morphisms the classical purity results for X/Y A finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der . Waerden’spurity for birational morphisms. h t a Introduction m [ In this paper π : X/S → Y/S denotes a dominant morphism locally of finite type of Nœtherian 3 integral S-schemes of relative dimension d . Let Ω be the sheaf of relative differentials, i.e. X/Y X/Y v 2 Ω =Coker(π∗(Ω )→Ω ), 7 X/Y Y/S X/S 8 and dually let C = Coker(dπ :T →T ) be the critical module of π, where dπ is the 5 X/Y X/S X/S→Y/S tangentmorphismofπ. The critical locus ofπ is the supportC of C , andthe branch locus B . π X/Y π 3 is the set of points x where the stalk Ω is not free; we abuse the terminology since B is the X/Y,x π 0 set oframificationpoints as defined in [20] only whenΩ is torsionandhence B =suppΩ . 0 X/Y π X/Y These two subsets of X, satisfying C ⊂ B , exert much control on the morphism π. If B = ∅, 1 π π π : thenπ issmooth(asdefinedin[20,Def. 17.3.1])ifitisflatandgenericallysmooth,andifmoreover v π is finite and Y is normal, then π is étale [3, Sec. 4]; see [21] for a discussion of the relation i X between the branch locus and the non-smoothness locus. If C = ∅, and π is either flat or Y/S is π r smooth, then tangentvector fields on Y lift (locally) to tangent vector fields on X,so accordingto a Zariski’slemma(CharX =0)themorphismπislocallyanalyticallytrivial. Itisthereforeanatural problem to find upper bounds on the codimensions of B and C , so that B = ∅ or C = ∅ can π π π π be controlled in low codimensions. The best situation is when codim C ≤1 and codim B ≤1 X π X π (whennonempty),andwesaythatC andB arepure(ofcodimension1),respectively. LetF (M) π π i denote the ith Fitting ideal of a module and the relative dimension d be defined as the Krull X/Y dimension of the generic fibre of π. If X/S and Y/S are smooth there is a duality relation F (C )=F (Ω ) (∗) i X/Y dX/Y+i X/Y 12000MathematicsSubjectClassification: Primary: 14A10,32C38;Secondary: 17B99(Secondary) 1 (Prop. 1.3), so in particular C =B in this case. A simple notable fact is that codim C ≤ 1 if π π X π the image of the tangent morphism Im(dπ) satisfies Serre’s condition (S ), and that this holds in 2 particularwhenX satisfies(S )andπ isgenericallyseparablyalgebraic;henceby(∗)codim B ≤ 2 X π 1 when X/S and Y/S are smooth. In general we shall see that C is pure “more often” than B π π (Th.3.5). OurmethodofestablishingpurityresultsforC andB isbyassumingagoodbehaviour π π of the modules Ω and Ω . Say that π is a differentially complete intersection morphisms X/S Y/S (d.c.i.) at a point x if the projective dimension p.d.Ω ≤ 1. This is inspired by a result due X/Y,x to Ferrand and Vasconcelos [13,31] that in the case of generically smooth domains over a field, which, when extended to a relative situation, states that locally complete intersection morphisms X/Y that are smooth at all the associated points in X are d.c.i., and that the converse holds if p.d. O <∞for eachpoint x∈X; we include a complete proof(Th. 2.1). In Section2 we O X,x Y,π(x) work out some basic results for d.c.i. morphisms, demonstrating that not only is the class of d.c.i. morphisms larger, for singular Y, than the class of locally complete intersection morphisms, but it is also in some respects easier to work with. This is exemplified by a more general base-change theorem (Th. 2.9) for locally complete intersection morphisms than in [5, 5.11], using a simpler argument. Recallingthatbase-changesX /Y aresmoothwhenX/Y is smooth,itisinterestingto 1 1 see that X /Y is d.c.i. when X/Y is d.c.i. and smooth at all points are are images of associated 1 1 points in X ; this also often implies that X /Y is a locally complete intersection. 1 1 1 In Theorem 3.5 we give general upper bounds of codim B(i) and codim C(i), where B(i) and X π X π π (i) (0) (0) C are higher order branch and critical loci (B = B and C = C are the most important π π π π π sets), defined by certain Fitting ideals, in terms of defect numbers of X/S and Y/S, the more precise the more assumptions on π are made. Let δ be the relative smoothness defect, χ (M) X/Y 2 a partial Euler characteristic, and β (M) the second Betti number of a module (see Section 3 for 1 precise definitions). We state a similar but somewhat more precise result: Theorem 3.8. Let π : X/S → Y/S be a generically smooth morphism of Nœtherian integral S-schemes such that Ω and Ω are coherent. X/S Y/S (1) codim+ B(i) ≤(d +i+1)(i+1+δ +χ (Ω )). X π X/Y Y/S 2 X/S In particular, if X/S is d.c.i., then codim+ B(i) ≤(d +i+1)(i+1+δ ). X π X/Y Y/S (2) Assume that X/S and Y/S are d.c.i., then codim+ B(i) ≤(d +i+1)(i+1+χ (Ω ))≤(d +i+1)(i+1+β (Ω )). X π X/Y 2 X/Y X/Y 1 X/Y (3) Assume that X/S is smooth and that each restriction to fibres X →Y , s∈S, is generically s s smooth, then codim+ B ≤δ +d . X π X/Y X/Y The proof of (1) results from a decomposition property for d.c.i. morphisms in Theorem 2.14 combined with a bound on the height of determinant ideals by Eagon and Northcott [14]. When X/Y itself is a d.c.i., then χ (Ω ) = 0, and the first inequality in (2) results in an inequality 2 X/Y due to Dolgachev (when i=0). The proof of (3) is based on a refinement of the Eagon-Northcott 2 bound by Eisenbud, Ulrich and Huneke [12]. Clearly, (1) is more interesting than (2) and (3) if we have more knowledge of the ramification of X/S and Y/S than the ramification of X/Y. One may reflect a moment over the usefulness of our relative setting, bearing in mind that the non-smoothness and branch loci are unions of the corresponding loci for morphisms of fibres X →Y , taking one point s in S at a time [20, Props 17.8.1, 17.8.2]. It is nevertheless natural to s s considerthe schemesoverS,since,forexample,ifS is definedoverafieldk itisnotnecessarythat X/k andY/k satisfyverystrongrequirements,saynecessaryto applyZariski-Nagata-Auslanderor Grothendieck purity (described below), as long as the fibres X and Y are sufficiently nice. s s In Section 4 we discuss the special case d = 0. On the one hand, if π : X → Y is a finite X/Y morphsim we have classical result by Zariski, Nagata and Auslander [4,30,33], stating that B π is pure when π is finite, X is normal, and Y is regular. This was generalised by Grothendieck, Faltings, Griffith, Cutkosky, and Kantorovitz [10,15,17,18,23], allowing some singularities in Y. On the other hand, for birational morphisms van der Waerden’s purity theorem states that B is π purewhenY isnormalandacertaincondition(W)issatisfied(e.g. Y isQ-factorial). Itistherefore naturaltoaskasin[20,Rem. 21.12.14,(v)],ifthetwotypesofpurity,oneforfiniteandanotherfor birationalmorphisms,canbe usedtogetherso thatone getspurity for genericallyfinite morphisms from the mere existence of a factorisation into a birational and a finite morphism X −→f Y′ −→g Y, where Y′ is the normalisation of Y in X. However, Y′ need not satisfy (W); more precisely, the complement of the branch locus B of the finite morphism need not be affine (see Remark 4.2). g We work out necessary conditions in Theorem 4.4 to ensure that this phenomenon does not occur. Moreover, we get other quite general positive results when π is only separably algebraic, which is thus less than what is required both in the Zariski-Nagata-Auslander purity theorem and van der Waerden’s purity theorem. The first is (1) in Theorem 3.8, implying codim B ≤ 1 when X/S X π d.c.i. and Y/S is smooth. However, it is often important to allow that either X/S or Y/S is not d.c.i.. First in arbitrary characteristic,Theorem 4.6 gives codim B ≤1, still assuming that X/S X π is d.c.i., together with a regularity condition at points of low height in X and Y, respectively, and thatthecanonicalmapπ∗(Ω )→Ω beinjective. WhenthebaseschemeS isdefinedoverthe Y/S X/S rational numbers we have Theorem 4.5, again ensuring codim B ≤ 1, which can be regarded as X π a generalisedrelative versionof the Zariski-Nagata-Auslanderpurity theoremin the sense that the conditionsonX andY areofasimilartype,namelythatY/S issmoothandX satisfies(S ),while 2 π is only generically algebraic and not necessarily finite. In general, even if π is finite, B need π not be pure of codimension 1 when Y/S is non-smooth and the bound on codimB will depend π on the type of singularities. For example, we include a simple argument for Cutkosky’s bound codim B ≤ 2 when π is finite, X is normal, and Y is a local complete intersection (Prop. 4.1); X π it is really a direct consequence of Grothendieck’s purity theorem. Our rather generalbounds that arise from Theorems 3.5 and 3.8 in terms of defect numbers of X/S and Y/S are interesting to compare to a bound by Faltings and Cutkosky in terms of the regularity defect of Y, where the latter is applicable only when π is finite. Not only is our bound easier to get and more general in that π need not be finite, in the finite case it improves the Faltings-Cutkosky bound when π is defined over some base S and the singularities of X and Y are fibered over S (see discussion after Proposition 4.1). Generalities: AllschemesareassumedtobeNœtherianandweconformtothenotationinEGA. The height ht(x) of a point x in X is the same as the Krull dimension of the local ring O , and X,x the dimension of X is dimX =sup{ ht(x) | x∈X}. A point x is a maximal point in a subset T of X if for each point y in the closure of x in T we have ht(x)≤ht(y), i.e., if x ∈T specialises to x 1 3 and ht(x ) ≤ ht(x), then x = x. Denote by Max(T) the set of maximal points of T, so Max(X) 1 1 consists of points of height 0. A property on X is generic if it holds for all points in Max(X). We assume that X is generically reduced, so k =O when ξ ∈Max(X) (so if the nilradicalof O X,ξ X,ξ X is non-zero, then all its associted points are embedded points in X). Put codim+ T =sup{ ht(x) | x∈Max(T)}, X codim− T =inf{ ht(x) | x∈Max(T)}, X so codim− T ≤ ht(x) ≤ codim+ T when x ∈ Max(T) (in the introduction we mean codim = X X codim+); one may call codim+ T and codim− T the upper and lower codimension of T in X, X X respectively. IfT isthe emptyset,putcodim+ T =−1andcodim− T =∞,sinceweareinterested X X in lower and higher bounds of codim± T, respectively. For a coherent O -module M we put X X depth M = inf{depthM | x ∈ T}. We define the relative dimension of a morphism locally of T x finite type π : X → Y at a point x ∈ X as the infimum of the dimension of the vector space of Kähler differentials at all maximal points ξ that specialise to x, i.e. d =inf{dim Ω | x∈ξ−,ξ ∈Max(X)}. X/Y,x kX,ξ X/Y,ξ Put d =inf{dim Ω | ξ ∈Max(X)}. If X/Y is generically smooth X, and the numbers X/Y kX,ξ X/Y,ξ d andd donotdependonthechoiceofmaximalpointξ ∈Max(X)thatspecialisestox, X/S,ξ Y/S,π(ξ) thend =d −d . Recallthatdim Ω =dim Ω isthesameasthe X/Y,x X/S,x Y/S,π(x) kX,ξ X/Y,ξ kX,ξ kξ/kπ(ξ) transcendence degree and dimension of a p-basis of k /k in characteristic 0 and p, respectively, ξ π(ξ) and these numbers are equal when the field extension is separable and finitely generated. Thus d = 0 when x ∈ ξ−, ξ ∈ Max(X), O = k and k /k is separably algebraic, and if X/Y,x X,ξ X,ξ X,ξ π(ξ) moreover X is integral, then d =tr.degk /k . X/Y,x X,ξ Y,π(ξ) 1 Critical scheme and branch scheme AssumethatX isaconnectedschemeandπ :X →Y bealocallyoffinitetypedominantmorphism. The first fundamental exact sequence of quasi-coherent O -modules X 0→Γ →π∗(Ω )−→p Ω →Ω →0. (1.1) X/Y/S Y/S X/S X/Y contains the kernel Γ of p, which is called the imperfection module of X/Y/S. Denoting the X/Y/S image by V we have two short exact sequences X/Y/S 0→Γ →π∗(Ω )→V →0, (1.2) X/Y/S Y/S X/Y/S 0→V →Ω →Ω →0. (1.3) X/Y/S X/S X/Y i p ConsiderachainofmorphismsX −→X −→Y →S andputπ =p◦i. Thereisanexactsequence r [19, Th. 20.6.17] 0→ΓX →Γ →Γ →i∗(Ω )→Ω →Ω →0, (1.4) Xr/Y/S X/Y/S X/Xr/S Xr/Y X/Y X/Xr where ΓX =Ker(i∗(p∗(Ω )))→i∗(Ω )). Xr/Y/S Y/S Xr/S 4 WewillstudytheFittingidealsF (Ω ),i≥0,definingtheithbranchschemeB(i) (Ka¨hler dX/Y+i X/Y π different, Jacobians), so there is a finite decreasing filtration of B π ⊂B(i) ⊂B(i−1) ⊂···⊂B0 =B . π π π π Here B is the branch scheme and its underlying topological space is the branch locus. π Remark 1.1. (1) If Y/S is smooth and X/Y is generically smooth, then Γ = 0, but X/Y/S Γ is normally non-zero, also in characteristic 0 if Y/S is non-smooth. Example: A = X/Y/S k[x,y]/(x2+y3),B =k[x′,y′]/(x′2+y′) and let A→B be defined by x′ =xy and y′ =y (a chartofthestricttransformwithrespecttotheblow-upoftheorigin). Thetorsionsubmodule of Ω is A(2ydx−3xdy) and Γ =B(2ydx−3xdy)⊂B⊗ Ω . A/k B/A/k A A/k (2) The two middle termes in (1.2) are quasi-coherent so Γ is quasi-coherent, and if Y/S X/Y/S is of finite type, then Γ is coherent, since X is a noetherian space. Also, if only X/Y X/Y/S is locally of finite type, then Γ is coherent; this is proven using the sequence (1.4)). X/Y/S Assume that X,Y,S are integral and that the fraction field of S is perfect. If γ is the X/Y/S generic rank of Γ , then d = tr.degk /k +γ (Cartier’s equality), and X/Y/S X/Y X,ξ Y,π(ξ) X/Y/S if X/Y is generically smooth, then d =tr.degk /k (see [19, §20.6]). X/Y X,ξ Y,π(ξ) (3) We have B(i) = X when i < 0 and B(i) = ∅ when i ≥ sup {β (Ω )}−d . Hence π π x 0 X/Y,x X/Y codim− B ≤ 1, always, and if X/Y is generically smooth, then B = B(0). If d = 0, X π π π X/Y then B =suppΩ , and B is the locus of points where X/Y is not formally unramified, π X/Y π while if d >0, then B is a subset of the non-smoothness locus of X/Y (if π is locally of X/Y π finite type and flat, then B is the non-smoothness locus. π ThedualofpinducesahomomorphismofO -modules,thetangentmorphism ofπ,dπ :T = Y X/S Hom (Ω ,O )→T where the O -module of ‘derivations from O to O ’ is OX X/S X X/S→Y/S X Y X TX/S→Y/S =HomOX(π∗(ΩY/S),OX)=Homπ−1(OY)(π−1(ΩY/S),OX), and is part of the exact sequence dπ 0→T →T −→T →C →0, (1.5) X/Y X/S X/S→Y/S X/Y whereC isthecriticalmodule ofπ. ThecriticalsetC =suppC isendowedwithastructure X/Y π X/Y of a scheme (still denoted C ), defined by the Fitting ideal F (C ); we say that π is submersive π 0 X/Y at a point x in X if x ∈/ C . Note that T = π∗(T ) when either π is flat or Ω is π X/S→Y/S Y/S Y/S locally free of finite rank. Let C(i) be the scheme that is defined by the Fitting ideal F (C ), giving π i π a finite decreasing filtration of the critical scheme C π ⊂C(i) ⊂···⊂C(1) ⊂C(0) =C . π π π π (i) Remark 1.2. If Ω is locally free it is straightforward to see that the space of B is given by X/S π a rank condition on the induced map of fibres of the map p, while if T is locally of finite rank, Y/S the space of C(i) is given by a rank condition on the induced map of fibres of the map dπ. π Proposition 1.3. If X/S and Y/S are generically smooth and B =∅, B =∅, then X/S Y/S F (C )=F (Ω ). i X/Y dX/Y+i X/Y 5 To be clear, note that the requirements in Proposition1.3 are satisfied when X/S and Y/S are smooth, and thus in this situation C(i) =B(i). The reason is that C and Ω are transposed π π X/Y X/Y modulesofoneanother,sofortheproofoneneedsarelationbetweentheFittingidealsofamodule and its transpose. Lemma 1.4. Let φ : G → G be a homomorphism of locally free O - modules of finite ranks g 1 2 X 1 and g , respectively. Let φ∗ :G∗ →G∗ be the homomorphism of dual modules. Then 2 2 1 F (Cokerφ)=F (Cokerφ∗). i g1−g2+i Proof. Fi(Cokerφ) is the image of the map ∧g2−iG1⊗OY (∧g2−iG2)∗ → OX induced by the map ∧g2−iφ:∧g2−iG1 →∧g2−iG2 and Fr(Cokerφ∗) is the image of the map ∧g1−rG∗2⊗OY ∧g1−rG1 → OX induced by the map ∧g1−rφ∗ :∧g1−rG∗2 →∧g1−rG∗1. When g2−i=g1−r, i.e. r =g1−g2+i we get a commutative diagram ∧g2−iG1⊗OY (∧g2−iG2)∗ //OX (cid:15)(cid:15) (∧g2−iG∗1)∗⊗OY ∧g2−iG∗2 //OX wheretheleftverticalhomomorphismexistsbecausetherearecanonicalmaps∧g2−iG1 →(∧g2−iG∗1)∗ and ∧g2−iG∗2 →(∧g2−iG2)∗, and the latter is an isomorphismbecause G2 is locally free (they both are isomorphisms since G also is locally free). 1 Proof of Proposition 1.3. The assumptions give that Ω and Ω are locally free of ranks X/S Y/S d and d , respectively, so dπ : G = T → G = π∗(T ) is a homomorphism of locally X/S Y/S 1 X/S 2 Y/S free O -modules, where Coker(dπ) = C and Coker(dπ∗) = Ω , so the result follows from X X/Y X/Y Lemma 1.4, noting that d =d −d . (cid:3) X/Y X/S Y/S Remark 1.5. RecallthatthekernelandcokernelofabidualitymorphismM →M∗∗ ofacoherent O -module M can be expressed using the transposed module D(M), locally defined up to local X projective equivalence by D(M)=Coker(φ∗), where φ is a local presentation F −→φ F →M →0. 1 0 Then we have the exact sequence 0→Ext1 (D(M),O )→M →M∗∗ →Ext2 (D(M),O )→0; (1.6) OX X OX X see [2]. Note also that when the projective dimension p.d.M ≤1 for each point x, then D(M) is x locally projectively equivalent to Ext1 (M, O ). OX X 2 Differentially complete intersections Ferrand and Vasconcelos [13,31] have shown that if X/k is a reduced scheme locally of finite type andgenericallysmooth(i.e. theresiduefieldsatallmaximalpointsareseparableoverk),thenX/k is a locally complete intersection if and only if the projective dimension p.d.Ω ≤ 1 at each X/k,x point x; see also [24, §9]. Because of this result there are two natural notions of “locally complete intersection morphisms”. The first is well-known in the case when π : X → Y is “smoothable”: there exists a locally defined factorisation X → Z → Y, where X/Z is a regular immersion and 6 Z/Y is formally smooth, i.e. the ideal of X in Z is locally defined by a regular sequence; this was furtherdevelopedin[5]togeneralmorphismsemploying“Cohen-factorisations”,provingthatthere isanalternativedefinitionbythevanishingofcertainAndré-Quillenhomologygroups. Wecontinue to call such morphisms locally complete intersection morphisms (l.c.i.), but we shall however have more use for a second possibility. Say that a dominant morphism π : X → Y is a differential complete intersection (d.c.i.) at a point x if p.d.Ω ≤1, and that π is a d.c.i. if it is d.c.i. at X/Y,x each point x; we then also write p.d.Ω ≤1. Let x be a specialisation of a point ξ in X. Since X/Y Ω =O ⊗ Ω , it is evident that a morphism is a d.c.i. at ξ if it is d.c.i. at x, hence X/Y,ξ X,ξ OX,x X/Y,x it suffices to check the closed points in X to see if a morphism is d.c.i. If the first syzygy of the quasi-coherentmodule Ω is coherentit is clearthat the set {x∈X | π is a d.c.i. at x} is open. X/Y Recall also that Ω =Ω (see [22] for a proof not using the fact that O →O X/Y,x OX,x/OY,π(x) X,x X,ξ is etale), so d.c.i. is a property of the morphism of local rings O →O . If π is smoothable Y,π(x) X,x and l.c.i. at x then it is l.c.i at ξ, but the proof of this assertion is not as immediate as for the d.c.i. property; for non-smoothable morphisms this localisation property for l.c.i. morphisms need not hold, see [5, 5.3, 5.12]. Theorem 2.1. (Ferrand, Vasconcelos) Let π : X/S → Y/S be morphism locally of finite type. Consider the conditions for a point x in X: (1) π is l.c.i. at x. (2) π is d.c.i. at x. If X/Y is smooth at all associated points in X then (1) ⇒ (2). If X/Y is generically smooth and p.d. O <∞ then (2)⇒(1). O X,x Y,π(x) Remark 2.2. Kunz gives a part of the proof in the above relative situation [24, Th. 9.2], but it seems that the possibility of embedded points is overlooked(flatness is included in his definition of locally complete intersection, but this does not rule out this possibility). We record a situation where no embedded points are present in X, so the above smoothness conditions can be expressed more concretely as X being geometrically reduced over the maximal points in Y. The proof is immediate. Lemma 2.3. If Y is Cohen-Macaulay and X/Y is l.c.i., then X is Cohen-Macaulay, and hence contains no embedded points. Theproofofthe followinglemmain[31]is perhapsalittle succinct,sowe include anargument. Lemma2.4. (Vasconcelos)LetJ beaproper idealofaNœtherianlocalringA,suchthatp.d. J < A ∞. If J/J2 = Γ⊕K where Γ is free of rank l over A/J, then J contains a regular sequence of length l, and if K 6=0 it contains a regular sequence of length l+1. Proof. Put B =A/J. Since p.d. J <∞, hence p.d. B <∞, and since A is local, B has a finite A A free resolution as A-module. Since Ann (B) = J 6= 0 by Auslander-Buchsbaum’s theorem [3] J A contains an A-regular element, so J 6⊂ P for each associated prime P of A. By prime avoidance there exists an element x ∈ J such that x 6∈ P for all associated primes (so again x is a regular element) and x ∈/ m J. The image x¯ of x for the projection J/J2 → Γ satisfies x¯ 6∈ m Γ, so it A B can be complemented to a basis {x¯ = x¯,x¯ ,...,x¯ } of the free B-module Γ, hence Bx¯ is a free 1 2 l summand of Γ⊂J/J2. Select x ∈J that project to x¯ , i=2,...,l. i i 7 Now put A∗ =A/(x) and J∗ =J/(x). Since x is A-regular it is also J-regular, so by [27, Lem. 2, §18] p.d.A∗J/xJ < ∞. Since x 6∈ mAJ it follows that the natural map J/xJ → J∗ splits (see proof of [Th 19.2, loc cit]). Therefore p.d.A∗J∗ ≤p.d.A∗J/xJ <∞. Since Bx¯ is a free summand of Γ and hence a free directsummand of J/J2, it followsthat J∗/(J∗)2 =J/(Ax+J2)∼=Γ∗⊕K∗, where Γ∗ is a free module of rank l−1, generated by x mod((x)+J2), i = 2,...,l. If K = 0 we i see by induction that I is generated by a regular sequence of length l. If K 6=0 againby induction it follows that I contains a regular sequence of length l+1. i Proof of Theorem 2.1. (1) ⇒ (2): There exists locally a factorisation X −→ X → Y where r X/X is a regular immersion and X /Y is smooth. Letting I be the ideal of X in X we get the r r r exact sequence 0→K →I/I2 →i∗(Ω )→Ω →0. Xr/Y X/Y SinceX/X isaregularimmersion,soI islocallygeneratedbyaregularsequence,theO -module r X,x I /I2 is free of some rank l. Note in passing, Ω being free of rank d it follows that x x Xr/Y,x Xr/Y,x rankΩ =l−d for each ξ ∈Max(X). Moreover,Ω is free when x is an associated kξ/kπ(ξ) Xr/Y X/Y,x point in X, implying that I /I2 =Γ⊕K , wheren Γ is free of rank l. Since I does not contain a x x x x regularsequenceoflengthr+1,Lemma2.4impliesK =0,andsinceK isasubmoduleofalocally x free O -module, K = 0. Since I/I2 and Ω are locally free it follows that p.d.Ω ≤ 1 at X Xr/Y X/Y,x each point x. i (2)⇒(1): There exists locally a factorisationX −→X →Y where X /Y is smooth and X/X r r r is a closed immersion. Consider the exact sequence 0→Γ →i∗(Ω )→Ω →0. X/Xr/Y Xr/Y X/Y Sincep.d.Ω ≤1ateachpointxandΩ isfree,itfollowsthatΓ isfree. Putting X/Y,x Xr/Y,x X/Xr/Y,x l=d −d , since X/Y and X /Y are generically smooth, rankΓ =l. Combining Xr/Y,x X/Y,x r X/Xr/Y with the previous exact sequence we get the split exact sequence 0→K →I /I2 →Γ →0, x x X/Xr/Y,x where K is torsion. Put A = O , J = I and Γ = Γ , so J/J2 = Γ⊕K, where Γ is Xr,x x X/Xr/Y,x A-free of rank l, and as the maximal length of an A- regular sequence in J satisfies depth A ≤ J dimA−dimO = l Lemma 2.4 implies K = 0; we only have to note that p.d. O < ∞ X,x O X,x implies p.d. O <∞ and hence p.d. J <∞. (cid:3) Y,π(y) A X,x A Remark 2.5. (1) If π is not generically smooth, then (1) does not imply (2) in Theorem 2.1. Example: A=k[x]/(x2) is l.c.i. over k, but p.d. Ω =p.d. k =∞. A A/k A (2) ForaregularbaseY,genericallysmoothd.c.i. morphismsarethe sameasgenericallysmooth locally complete intersection morphisms, but if p.d. O =∞ for some point x, then O X,x Y,π(x) (2) does not imply (1) in Theorem 2.1. Example: A=k[t]/(t2), k is a field, R=A[x,y] and I =(t+x,t+yx2). Put B =R/I and consider the natural map A→B. Then (a) In the ring B we have txy(t+x) = t(x2y+t) and I cannot be generated by a regular sequence. (b) The B-module I/I2 is free of rank 2, so by (a) and [32] p.d. I =∞. R (c) The sequence 0→I/I2 →B⊗ Ω →Ω = k[t,x,y] →0 is exact also to the left. R R/A B/A (t2,x2,y) 8 Therefore p.d.Ω ≤1 while A→B is not l.c.i. See also Theorem 2.14, (3-4). B/A Lemma 2.6. Let π : X → Y be a locally of finite type morphism that is smooth at all associated points in X. Assume either: (1) X/S and Y/S are smooth. (2) X and Y are regular schemes. Then π is d.c.i. (1) was first observed by Dolgachev [11]; in Theorem 2.14 we will give another necessary condition for X/Y to be d.c.i. Clearly, (1) ⇒ (2) when X and Y are geometrically regular over a field. Proof. (1): Since π∗(Ω ) is locally free and π is smooth all associated points, implying that the Y/S imperfection module Γ is 0 at all associated points, hence Γ =0 in the exact sequence X/Y/S X/Y/S (1.1). Since moreover Ω is locally free, it follows that p.d.Ω ≤ 1 at each point x. (2): X/S X/Y,x i p There exists locally a factorisation X −→ X −→Y, where i is a closed immersion and p is smooth. r Since p is smooth and Y is regular, it follows that X is regular. Since X is regular and i is a r closed immersion, it must be a regular immersion. Therefore π is l.c.i., hence by Theorem 2.1 π is d.c.i. We give necessary conditions to conclude that a morphism is d.c.i. when all its fibres are d.c.i. Proposition 2.7. Let π :X →Y be a flat dominant morphism locally of finite type of Nœtherian schemes, which is smooth at all associated points of X. Assume either of the conditions: (1) Ω is Y-flat. X/Y (2) each fibre X /k ,y ∈Y, is generically smooth (i.e. generically geometrically reduced). y Y,y If the fibre X /k is d.c.i., then π is d.c.i. at each point x in X ⊂ X. Hence if each (closed) y Y,y y fibre X is d.c.i., then π is d.c.i. y Next easy lemma is standard, and a similar assertionholds for higher bounds on the projective dimension. Lemma 2.8. Let π :X →Y be a flat morphism of schemes and M be a coherent O -module, flat X over Y. Let x be a point in X, X be the fibre over y =π(x), and M the restriction of M to X . y Xy y If p.d.M ≤1, then p.d.M ≤1. Xy,x x Proof. Locally there exists a presentation 0 → L → F → M → 0 where F is locally free of finite rank. It suffices to see that L is free. Applying k ⊗ · to the exact sequence, by assumption x Y,y OY,y k ⊗ L is free over O , since M is flat over O and p.d. k ⊗ M ≤ 1. SYel,eyctinOgY,ya baxsis kY,y⊗OY,yOXXny,x,x∼=kY,y⊗OxY,yLx, arisingfrYo,my a homomOoXrpyh,xismY,yu:OOYXn,y,x →xLx of O -modules, so u is surjective by Nakayama’s lemma. Since M and O are flat, hence L is X,x x X,x x flat over O , we conclude that u is an isomorphism [27, Th. 22.5]. Y,y 9 Proof of Proposition 2.7. (1): ThisfollowsimmediatelyfromLemma2.8,notingthatk ⊗ Y,y OY,y Ω =Ω . X/Y,x Xy/kY,y (2): We can assume that X/Y is a subscheme of a smooth scheme Xr/Y so there is the short exact sequence 0→Λ→O ⊗ Ω →Ω →0, X OXr Xr/Y X/Y and the assertion is that Λ is free over O when x ∈ π−1(y). Since Xr/Y is smooth it follows x X,x that the two terms to the right are coherent, so Λ is of finite type. Let I be the defining ideal of x X inX (defined locally). We havea surjectivemapI /I2 −d→x Λ →0. Since p.d.(Ω )≤1 r x x x Xy/kY,y,x for each point x∈X if follows by Theorem 2.1 that X /k is l.c.i., since X /k is generically y y Y,y y Y,y smooth;hencesinceπ isflat,I isgeneratedbyaregularsequencesoI /I2 isfreeoverO . Since x x x X,x X/Y is smooth at each associated point ξ in X, and therefore d is injective, it follows that d is ξ x injective; hence Λ =I /I2 is free. Therefore p.d.Ω ≤1. (cid:3) x x x X/Y,x The class of d.c.i. morphisms behaves almost as well under base-change as the class of smooth morphisms. Consider a base change diagram over some scheme S: j X1 // X (BC) π1 π (cid:15)(cid:15) (cid:15)(cid:15) Y // Y, 1 where X =X × Y . 1 Y 1 Theorem 2.9. Let π :X/S →Y/S be a dominant morphism locally of finite type where X/S and Y/S are Nœtherian, and assume that π is smooth at all points in j(X ) ⊂ X that are images of 1 associated points in X . If π is d.c.i. then π :X →Y is d.c.i. 1 1 1 1 Note that when π is flat, then π is smooth at the associated points of j(X )⊂X if and only if 1 π is smooth at the associated points of X [19, Th 19.7.1]. 1 1 Lemma 2.10. Let j :X →Y be a morphism of schemes and M a coherent O -module satisfying Y p.d.M ≤ 1 at each point y in the image j(X). Assume also that M is flat over O when y is y y Y,y an associated point in j(X). Then p.d.j∗(M) ≤1 at each point x in X. x Proof. Let x be a point inX andset B =O and A=O . If 0→F1 →F0 →M →0 is X,x Y,π(x) j(x) exact, where F ,F are free, then 0→Tor1(B,M )→B⊗ F1 →B⊗ F0 −→h j∗(M) →0 is 1 0 A j(x) A A x exact. ByassumptionthesupportofthaB-moduleTor1(B,M )doesnotcontainanyassociated A j(x) point of B and B⊗ F1 is free; therefore Tor1(B,M )=0, implying p.d.j∗(M) ≤1. A A j(x) x Proof of Theorem 2.9. This follows from Lemma 2.10, noting that Ω =j∗(Ω ). (cid:3) X1/Y1 X/Y Theorem 2.1 immediately implies the following corollary to Theorem 2.9: Corollary 2.11. Make the same assumptions as in Theorem 2.9, and assume moreover that p.d. O <∞ OY1,π(x) X1,x for all x in X (e.g. Y is regular). Then π is l.c.i.. 1 1 1 10

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