Fibers of rational maps and Rees algebras of their base ideals
aa r X i v : . [ m a t h . A C ] J u l Fibers of rational maps and Rees algebras of theirbase ideals
Tran Quang Hoa ∗ and Ho Vu Ngoc Phuong † University of Education, Hue University, 34 Le Loi St., Hue, Vietnam. University of Sciences, Hue University, 77 Nguyen Hue St., Hue, Vietnam.
Abstract
We consider a rational map φ : P mk P nk that is a parameterization of an m -dimensional variety. Our main goal is to study the ( m − φ in relation to the m -th local cohomology modules of the Rees algebra of its baseideal. Keywords : Approximation complexes, base ideals, fibers of rational maps, parameteriza-tions, Rees algebras
Let k be a field and φ : P mk P nk be a rational map. Such a map φ is defined by homoge-neous polynomials f , . . . , f n , of the same degree d, in a standard graded polynomial ring R = k [ X , . . . , X m ] , such that gcd( f , . . . , f n ) = 1 . The ideal I of R generated by thesepolynomials is called the base ideal of φ . The scheme B := Proj( R/I ) ⊂ P mk is called the base locus of φ . Let B = k [ T , . . . , T n ] be the homogeneous coordinate ring of P nk . Themap φ corresponds to the k -algebra homomorphism ϕ : B −→ R, which sends each T i to f i . Then, the kernel of this homomorphism defines the closed image S of φ. In otherwords, after degree renormalization, k [ f , . . . , f n ] ≃ B/ Ker( ϕ ) is the homogeneous coor-dinate ring of S . The minimal set of generators of Ker( ϕ ) is called its implicit equations and the implicitization problem is to find these implicit equations.The implicitization problem has been of increasing interest to commutative algebraistsand algebraic geometers due to its applications in Computer Aided Geometric Design asexplained by Cox [1].We blow up the base locus of φ and obtain the following commutative diagramΓ (cid:31) (cid:127) / / π (cid:15) (cid:15) P mk × P nkπ (cid:15) (cid:15) P mk φ / / ❴❴❴❴❴❴❴ P nk . The variety Γ is the blow-up of P mk along B , and it is also the Zariski closure of the graphof φ in P mk × P nk . Moreover, Γ is the geometric version of the Rees algebra R I of I, i.e., ∗ Corresponding Author: [email protected] † [email protected] R I ) = Γ . As R I is the graded domain defining Γ , the projection π (Γ) = S isdefined by the graded domain R I ∩ k [ T , . . . , T n ], and we can thus obtain the implicitequations of S from the defining equations of R I . Besides the computation of implicit representations of parameterizations, in geomet-ric modeling it is of vital importance to have a detailed knowledge of the geometry ofthe object and of the parametric representation one is working with. The question ofhow many times is the same point being painted (i.e., corresponds to distinct values ofparameter) depends not only on the variety itself but also on the parameterization. It isof interest for applications to determine the singularities of the parameterizations. Themain goal of this paper is to study the fibers of parameterizations in relation to the Reesand symmetric algebras of their base ideals. More precisely, we set π := π | Γ : Γ −→ P nk . For every closed point y ∈ P nk , we will denote its residue field by k ( y ). If k is assumed tobe algebraically closed, then k ( y ) ≃ k. The fiber of π at y ∈ P nk is the subscheme π − ( y ) = Proj( R I ⊗ B k ( y )) ⊂ P mk ( y ) ≃ P mk . Suppose that m ≥ φ is generically finite onto its image. Then, the set Y m − = { y ∈ P nk | dim π − ( y ) = m − } consists of only a finite number of points in P nk . For each y ∈ Y m − , the fiber of π at y is an ( m − P mk and thus the unmixed component ofmaximal dimension is defined by a homogeneous polynomial h y ∈ R. One of the interestingproblems is to establish an upper bound for P y ∈Y m − deg( h y ) in terms of d . This problemwas studied in [2, 3].The paper is organized as follows. In Section 2, we study the structure of Y m − . Someresults in this section were proved in [2]. The main result of this section is Theorem 2.5that gives an upper bound for P y ∈Y m − deg( h y ) by the initial degree of certain symbolicpowers of its base ideal. This is a generalization of [3, Proposition 1] where the firstauthor only proved this result for parameterizations of surfaces φ : P k P k under theassumption that the base locus B is locally a complete intersection. More precisely, wehave the following. Theorem
If there exists an integer s such that ν = indeg(( I s ) sat ) < sd , then X y ∈Y m − deg( h y ) ≤ ν < sd. In particular, if indeg( I sat ) < d , then P y ∈Y m − deg( h y ) < d. In Section 3, we study the part of graded m in X i of the m -th local cohomology modulesof the Rees algebra with respect to the homogeneous maximal ideal m = ( X , . . . , X m ) N = H m m ( R I ) ( − m, ∗ ) = ⊕ s ≥ H m m ( I s ) sd − m . The main result of this section is the following.
Theorem (Theorem 3.2)
We have that N is a finitely generated B -module satisfying B ( N ) = Y m − and dim( N ) = 1 . (ii) deg( N ) = P y ∈Y m − (cid:0) deg( h y )+ m − m (cid:1) . In the last section, we treat the case of parameterization φ : P k P k of surfaces. Weestablish a bound for the Castelnuovo-Mumford regularity and the degree of the B -module N = ⊕ s ≥ H m ( I s ) sd − , see Corollary 4.2 and Proposition 4.3. Proposition
Assume B = Proj( R/I ) is locally a complete intersection. Then reg( N ) ≤ n and deg( N ) ≤ (cid:18) n + 23 (cid:19) , where n = dim k H m ( R/I ) d − . Moreover, if indeg( I sat ) = d , then d ≤ n ≤ d ( d − . φ : P mk P nk Let n ≥ m ≥ R = k [ X , . . . , X m ] be the standard graded polynomialring over an algebraically closed field k . Denote the homogeneous maximal ideal of R by m = ( X , . . . , X m ). Suppose we are given an integer d ≥ n + 1 homogeneouspolynomials f , . . . , f n ∈ R d , not all zero. We may further assume that gcd( f , . . . , f n ) = 1 , replacing the f ′ i s by their quotient by the greatest common divisor of f , . . . , f n if needed;hence, the ideal I of R generated by these polynomials is of codimension at least two. Set B := Proj( R/I ) ⊆ P mk := Proj( R ) and consider the rational map φ : P mk − P nk x ( f ( x ) : · · · : f n ( x ))whose closed image is the subvariety S in P nk . In this paper, we always assume that φ isgenerically finite onto its image, or equivalently that the closed image S is of dimension m. In this case, we say that φ is a parameterization of the m -dimensional variety S .Let Γ ⊂ P mk × P nk be the graph of φ : P mk \ B −→ P nk and Γ be the Zariski closure ofΓ . We have the following diagram Γ π (cid:15) (cid:15) (cid:31) (cid:127) / / P mk × P nkπ (cid:15) (cid:15) P mk φ / / ❴❴❴❴ P nk where the maps π and π are the canonical projections. One has S = π (Γ ) = π (Γ) , where the bar denotes the Zariski closure. Furthermore, Γ is the irreducible subschemeof P mk × P nk defined by the Rees algebra R I := Rees R ( I ) = ⊕ s ≥ I s . P nk by B := k [ T , . . . , T n ]. Set S := R ⊗ k B = R [ T , . . . , T n ]with the grading deg( X i ) = (1 ,
0) and deg( T j ) = (0 ,
1) for all i = 0 , . . . , m and j =0 , . . . , n . The natural bi-graded morphism of k -algebras α : S −→ R I = ⊕ s ≥ I ( d ) s = ⊕ s ≥ I s ( sd ) T i f i is onto and corresponds to the embedding Γ ⊂ P mk × P nk .Let P be the kernel of α . Then, it is a bi-homogeneous ideal of S , and the part ofdegree one of P in T i , denoted by P = P ( ∗ , , is the module of syzygies of the Ia T + · · · + a n T n ∈ P ⇐⇒ a f + · · · + a n f n = 0 . Set S I := Sym R ( I ) for the symmetric algebra of I . The natural bi-graded epimorphisms S −→ S/ ( P ) ≃ S I and δ : S I ≃ S/ ( P ) −→ S/ P ≃ R I correspond to the embeddings of schemes Γ ⊂ V ⊂ P mk × P nk , where V is the projectivescheme defined by S I .Let K be the kernel of δ, one has the following exact sequence0 −→ K −→ S I −→ R I −→ . Notice that the module K is supported in B because I is locally trivial outside B .As the construction of symmetric and Rees algebras commutes with localization, andboth algebras are the quotient of a polynomial extension of the base ring by the Koszulsyzygies on a minimal set of generators in the case of a complete intersection ideal, itfollows that Γ and V coincide on ( P mk \ X ) × P nk , where X is the (possibly empty) set ofpoints where B is not locally a complete intersection.Now we set π := π | Γ : Γ −→ P nk . For every closed point y ∈ P nk , we will denote itsresidue field by k ( y ), that is, k ( y ) = B p / p B p , where p is the defining prime ideal of y. As k is algebraically closed, k ( y ) ≃ k. The fiber of π at y ∈ P nk is the subscheme π − ( y ) = Proj( R I ⊗ B k ( y )) ⊂ P mk ( y ) ≃ P mk . Let 0 ≤ ℓ ≤ m, we define Y ℓ = { y ∈ P nk | dim π − ( y ) = ℓ } ⊂ P nk . Our goal is to study the structure of Y ℓ . Firstly, we have the following.
Lemma 2.1 [2, Lemma 3.1]
Let φ : P mk P nk be a parameterization of m -dimensionalvariety and Γ be the closure of the graph of φ. Consider the projection π : Γ −→ P nk . Then dim Y ℓ + ℓ ≤ m. Furthermore, this inequality is strict for any ℓ > . As a consequence, π has no m -dimensional fibers and only has a finite number of ( m − -dimensional fibers. π are defined by the specialization of the Rees algebra. However, Reesalgebras are difficult to study. Fortunately, the symmetric algebra of I is easier to under-stand than R I , and the fibers of π are closely related to the fibers of π ′ := π | V : V −→ P nk . Recall that for any closed point y ∈ P nk , the fiber of π ′ at y is the subscheme π ′− ( y ) = Proj( S I ⊗ B k ( y )) ⊂ P mk ( y ) ≃ P mk . The next result gives a relation between fibers of π and π ′ – recall that X is the(possible empty) set of points where B is not locally a complete intersection. Lemma 2.2 [2, Lemma 3.2]
The fibers of π and π ′ agree outside X, hence they have thesame ( m − -dimensional fibers. The next result is a generalization of [4, Lemma 10] that gives the structure of theunmixed part of a ( m − π. Note that our result does not needthe assumption that B is locally a complete intersection as in [4], thanks to Lemma 2.2.Recall that the saturation of an ideal J of R is defined by J sat := J : R ( m ) ∞ . Lemma 2.3 [2, Lemma 3.3] Assume y = ( p : · · · : p n ) ∈ Y m − such that p i = 1 . Thenthe unmixed part of the fiber π − ( y ) is defined by h y = gcd( f − p f i , . . . , f n − p n f i ) . Furthermore, if f j − p j f i = h y g j for all j = i , then I = ( f i ) + h y ( g , . . . , g i − , g i +1 , . . . , g n ) and I sat ⊂ ( f i , h y ) . Remark 2.4
The above lemma shows that the ( m − π can onlyoccur when B 6 = ∅ as B ⊃ V ( f i , h y ) . It also shows that d deg( h y ) ≤ deg( B ) , if there is a ( m − h y . As a consequence,deg( h y ) < d for any y ∈ Y m − . By Lemma 2.1, π only has a finite number of ( m − R -module M is definedby indeg( M ) := inf { n ∈ Z | M n = 0 } with convention that sup ∅ = + ∞ . Theorem 2.5
If there exists an integer s ≥ such that ν = indeg(( I s ) sat ) < sd , then X y ∈Y m − deg( h y ) ≤ ν < sd. In particular, if indeg( I sat ) < d , then X y ∈Y m − deg( h y ) < d. roof. As Y m − is finite, by Lemma 2.3, there exists a homogeneous polynomial f ∈ I ofdegree d such that, for any y ∈ Y m − ,I = ( f ) + h y ( g y , . . . , g ny ) and I sat ⊂ ( f, h y )for some g y , . . . , g ny ∈ R . Since ( f, h y ) is a complete intersection ideal, it follows from [5,Appendix 6, Lemma 5] that ( f, h y ) s is unmixed, hence saturated for every integer s ≥ y ∈ Y m − ,( I s ) sat ⊂ (( I sat ) s ) sat ⊂ (( f, h y ) s ) sat = ( f, h y ) s = ( f s , f s − h y , . . . , h sy ) . Now, let 0 = F ∈ ( I s ) sat such that deg( F ) = ν < sd , then h y is a divisor of F .Moreover, if y = y ′ in Y m − , then gcd( h y , h y ′ ) = 1 . We deduce that Y y ∈Y m − h y | F which gives X y ∈Y m − deg( h y ) ≤ deg( F ) = ν < sd. Remark 2.6
In the case where φ : P k P k is a parameterization of surfaces. In [3],the first author showed that if B is locally a complete intersection of dimension zero, then X y ∈Y deg( h y ) ≤ ( d = 3 , (cid:4) d (cid:5) d − d ≥ . Example 2.7
Consider the parameterization φ : P k P k of surface given by f = X X ( X − X )( X + X )( X − X ) f = X X ( X − X )( X + X )( X − X ) f = X X ( X − X )( X + X )( X − X ) f = X X ( X − X )( X + X )( X − X ) . Using
Macaulay2 [6], it is easy to see that I = I sat and indeg(( I ) sat ) = 8 < . . Furthermore, I admits a free resolution0 / / R ( − ⊕ R ( − M / / R ( − / / R / / R/I / / , where matrix M is given by − X X − X )( X + X )( X − X )0 − X − ( X − X )( X + X )( X − X ) X X . Thus, we obtain Y = { p , p , p , p , p , p , p , p } with p = (0 : 0 : 0 : 1) h p = X p = (0 : 0 : 1 : 0) h p = X p = (0 : 1 : 0 : 1) h p = X − X p = (0 : − h p = X + X p = (0 : 2 : 0 : 1) h p = X − X p = (1 : 0 : 1 : 0) h p = X − X p = ( − h p = X + X p = (2 : 0 : 1 : 0) h p = X − X . Consequently, we have X y ∈Y deg( h y ) = 8 = indeg(( I ) sat ) . Local cohomology of Rees algebras of the base idealof parameterizations
Let φ : P mk P nk be a parameterization of m -dimensional variety. Let R = k [ X , . . . , X m ]and B = k [ T , . . . , T n ] be the homogeneous coordinate ring of P mk and P nk , respectively.For every closed point y ∈ P nk , the fiber of π at y is the subscheme π − ( y ) = Proj( R I ⊗ B k ( y )) ⊂ P mk ( y ) ≃ P mk and we are interested in studying the set Y m − = { y ∈ P nk | dim π − ( y ) = m − } . We now consider the B -module M µ = H m m ( R I ) ( µ, ∗ ) = ⊕ s ≥ H m m ( I s ) µ + sd , where m = ( X , . . . , X m ) is the homogeneous maximal ideal of R . By [7, Theorem 2.1], M µ is a finitely generated B -module for all µ ∈ Z . The following result gives a relationbetween the support of M µ and Y m − . For each y ∈ P nk = Proj( B ), we can see y as ahomogeneous prime ideal of B. Proposition 3.1
One has
Supp B ( M µ ) = { y ∈ Y m − | deg( π − ( y )) ≥ µ + m + 1 } . Proof. As k is algebraically closed, we have π − ( y ) = Proj( R I ⊗ B k ( y )) ⊂ P mk ( y ) ≃ P mk . Therefore, the homogeneous coordinate ring of π − ( y ) is R I ⊗ B k ( y ) ≃ R/J, where J is a satured ideal of R depending on y. Let y ∈ Y m − . As dim π − ( y ) = m − R I ⊗ B k ( y )) = dim R/J = m. Since dim R = m + 1, there exists a homogeneous polynomial f of degree d f such that J = ( f ) J ′ , with codim( J ′ ) ≥ . Notice that f is exactly the defining equation of unmixedpart of π − ( y ) . Consider the exact sequence0 −→ ( f ) /J −→ R/J −→ R/ ( f ) −→ H m m (( f ) /J ) −→ H m m ( R/J ) −→ H m m ( R/ ( f )) −→ H m +1 m (( f ) /J ) = 0 , since codim( J ′ ) ≥
2, hence ( f ) /J is of dimension at most m − . It follows from the aboveexact sequence that H m m ( R I ⊗ B k ( y )) ≃ H m m ( R/J ) ≃ H m m ( R/ ( f )) . (3.1)7e consider the following exact sequence0 / / R [ − d f ] × f / / R / / R/ ( f ) / / H m m ( R ) / / H m m ( R/ ( f )) / / H m +1 m ( R [ − d f ]) × f / / H m +1 m ( R ) / / . In degree µ , one has the following exact sequence0 −→ H m m ( R/ ( f )) µ −→ H m +1 m ( R ) µ − d f −→ H m +1 m ( R ) µ −→ . (3.2)On the other hand, H m +1 m ( R ) ≃ ( X · · · X m ) − k [ X − , . . . , X − m ] , hence H m +1 m ( R ) µ ≃ ( R ∗ ) µ + m +1 := Homgr R ( R − µ − m − , k ) . It follows that H m +1 m ( R ) µ = 0 for all µ > − m − H m +1 m ( R ) µ = 0 for any µ ≤ − m − H m m ( R/ ( f )) µ = ( µ > d f − m − R/ ( f )) ∗ µ − d f + m +1 = 0 if µ ≤ d f − m − . (3.3)By definition, M µ is a graded B -module and Supp B ( M µ ) ⊂ Proj( B ) . Now let p ∈ Proj( B ), we have p ∈ Supp B ( M µ ) ⇐⇒ M µ ⊗ B B p = 0 ⇐⇒ M µ ⊗ B B p ⊗ B ( B/ p ) = 0 ⇐⇒ H m m ( R I ) ( µ, ∗ ) ⊗ B k ( p ) = 0 ⇐⇒ H m m ( R I ⊗ B k ( p )) ( µ, ∗ ) = 0 . In particular, H m m ( R I ⊗ B k ( p )) = 0, hence dim( R I ⊗ B k ( p )) = m which shows that p ∈ Y m − . It follows from (3.1) and (3.3) that deg( π − ( p )) ≥ µ + m + 1 . In particular, if µ = − m , then the finitely generated B -module N = ⊕ s ≥ H m m ( I s ) sd − m satisfies Supp B ( N ) = Y m − by Proposition 3.1. Furthermore, we have the following. Theorem 3.2
Let N be the finitely generated B -module as above. Then (i) dim( N ) = 1 . (ii) deg( N ) = P y ∈Y m − (cid:0) deg( h y )+ m − m (cid:1) . roof. Let y = ( p : p : · · · : p n ) ∈ Y m − . Without loss of generality, we can assume that p = 1 . Hence, p = ( T − p T , . . . , T n − p n T ) ⊂ B is the defining ideal of y . For any f ∈ B , we have f = g ( T − p T ) + · · · + g n ( T n − p n T ) + v for some v ∈ k [ T ] . It follows that f + p = v + p . This implies that B/ p ≃ k [ T ] . Therefore,dim( B/ p ) = 1 for any p ∈ Y m − and thus, dim( N ) = max p ∈ Supp B ( N ) dim( B/ p ) = 1which shows (i). We now prove for item (ii). It was known that HP N ( s ) = HF N ( s ) = dim k N s = dim k H m m ( I s ) sd − m for all s ≫
0, where HP N and HF N is the Hilbert polynomial and the Hilbert functionof N , respectively. As dim N = 1, the Hilbert polynomial of N is constant which is equalto deg( N ) . On the other hand,deg( N ) = X dim( B/ p )=1 length B p ( N p ) . deg( B/ p ) . We proved that B/ p ≃ k [ T ] , hence dim( B/ p ) = 1 and deg( B/ p ) = 1 for the definingideal p of y ∈ Y m − . Thereforedeg( N ) = X y ∈Y m − length B p ( N p ) . As N p is an Artinian B p -module and dim k ( B/ p ) s = dim k ( k [ T ]) s = 1 for any s ≥
0, onehas length B p ( N p ) = dim k ( N ⊗ B B p )= X s dim k ( N ⊗ B B p ) s = X s dim k ( H m m ( R I ) ⊗ B B p ) ( − m,s ) = X s dim k ( H m m ( R I ) ⊗ B B p ) ( − m,s ) . dim k ( B/ p ) s = X s dim k H m m ( R I ⊗ B B p ⊗ B B/ p ) ( − m,s ) = X s dim k H m m ( R I ⊗ B k ( p )) ( − m,s )(3.1) = dim k H m m ( R/ ( f )) − m (3.3) = dim k ( R/ ( f )) d f − = dim k R d f − since deg( f ) = d f = (cid:18) d f + m − m (cid:19) .
9t follows that deg( N ) = X y ∈Y m − (cid:18) deg( h y ) + m − m (cid:19) . φ : P k P k of surfaces In this section, we consider a parameterization φ : P k P k of surface defined by fourhomogeneous polynomials f , . . . , f ∈ R = k [ X , X , X ] of the same degree d such thatgcd( f , . . . , f ) = 1. Denote the homogeneous maximal ideal of R by m = ( X , X , X ).From now on we assume that B is locally a complete intersection. Under this hypoth-esis, the module K in the exact sequence0 −→ K −→ S I −→ R I −→ m S . Hence, H i m ( K ) = 0 for any i ≥ . The above exact sequence deducesthat H i m ( S I ) ≃ H i m ( R I ) , ∀ i ≥ . Let B = k [ T , . . . , T ] be the homogeneous coordinate ring of P k . It follows from Theo-rem 3.2 that the finitely generated B -module N := ⊕ s ≥ H m ( I s ) sd − = H m ( R I ) ( − , ∗ ) ≃ H m ( S I ) ( − , ∗ ) satisfying dim( N ) = 1 andSupp B ( N ) = Y = { y ∈ P k | dim π − ( y ) = 1 } . Furthermore, X y ∈Y (cid:18) deg( h y ) + 12 (cid:19) = deg( N ) = dim k H m ( I s ) sd − for s ≥ reg( N ) + 1 , where reg( N ) is the Castelnuove-Mumford regularity of N . Thus, itis useful to establish the bounds for deg( N ) and reg( N ).Let K • := K • ( f ; R ) and Z • := Z • ( f ; R ) be the Koszul complex and the module ofcycles associated to the sequence f := f , . . . , f with coefficients in R , respectively. Sincethe ideal I = ( f ) is homogeneous, these modules inherit a natural structure of graded R -modules. Let Z • be the approximation complex associated to I . The approximationcomplexes were introduced by Herzog, Simis and Vasconcelos in [8] to study the Rees andsymmetric algebras of ideals. By definition Z q = Z q [ qd ] ⊗ R R [ T , . . . , T ]( − q )for all q = 0 , . . . , X i ) = (1 ,
0) and deg( T i ) = (0 , Z • ) : 0 / / Z v / / Z v / / Z v / / Z = R [ T , . . . , T ] / / v ( a , a , a , a ) = a T + · · · + a T . As B is locally a complete intersection, thecomplex ( Z • ) is acyclic and is a resolution of H ( Z • ) ≃ S I , see [9, Theorem 4].10 roposition 4.1 Assume B is locally a complete intersection. Then N admits a finiterepresentation of free B -modules B ( − m −→ B ( − n −→ N −→ , where n = dim k H m ( R/I ) d − and m = dim k H m ( Z ) d − . Proof.
We consider the two spectral sequences associated to the double complex C • m ( Z • ) , where C • m ( M ) denotes the ˇCech complex on M relatively to the ideal m . Since ( Z • ) isacyclic, one of them abuts at step two with: ∞ h E pq = h E pq = ( H p m ( S I ) for q = 00 for q = 0 . The other one gives at step one: v E pq = H p m ( Z q ) = H p m ( Z q )[ qd ] ⊗ R R [ T , . . . , T ]( − q ) = H p m ( Z q )[ qd ] ⊗ k B ( − q ) . By [9, Lemma 1], H p m ( Z q ) = 0 for p = 0 , Z ≃ R [ − d ] and Z = R .Therefore, the first page of the vertical spectral sequence has only two nonzero lines / / H m ( Z )[2 d ] ⊗ k B ( − / / H m ( Z )[ d ] ⊗ k B ( − / / H m ( Z )[3 d ] ⊗ k B ( − / / H m ( Z )[2 d ] ⊗ k B ( − / / H m ( Z )[ d ] ⊗ k B ( − / / H m ( Z ) ⊗ k B. In bi-degree ( − , ∗ ), we have H m ( Z ) − ⊗ k B = H m ( R ) − ⊗ k B = 0 . Therefore, we obtainthe complex ( C • ) of free B -modules0 / / B ( − l / / B ( − m / / B ( − n / / .C C C Notice that n = dim k H m ( Z ) d − = dim k H m ( I ) d − = dim k H m ( R/I ) d − . It remains to show that H ( C • ) = N. It is easy to see that ∞ v E pq = v E pq unless p = q = 3 or p = 2 , q = 1 . Therefore, M p − q =2 ∞ v E pq = v E = H m ( S I ) = M p − q =2 ∞ h E pq , in other words, H ( C • ) = H m ( S I ) ( − , ∗ ) = H m ( R I ) ( − , ∗ ) = N. We now establish a bound for the Castelnuovo-Mumford regularity and the degree of B -module N in terms of n = dim k H m ( R/I ) d − as follows. Corollary 4.2
Suppose B is locally a complete intersection. Then reg( N ) ≤ n and deg( N ) ≤ (cid:18) n + 23 (cid:19) . roof. As dim B = 4, hence codim( N ) = 3 and by Proposition 4.1, N admits a finiterepresentation B ( − m −→ B ( − n −→ N −→ . The corollary follows from [10, Corollaries 2.4 and 3.4].Theorem 2.5 shows that if indeg( I sat ) < d , then X y ∈Y deg( h y ) < d. Hence, the delicate case is when the ideal I satisfies indeg( I sat ) = indeg( I ) = d. In thiscase, the first author in [3] established an upper bound for n = dim k H m ( R/I ) d − in termsof d as follows. Proposition 4.3
Assume B is locally a complete intersection and indeg( I sat ) = d . Then d ( d + 1) ≤ deg( B ) ≤ d − d + 3 and d ≤ n = deg( B ) − d ( d − ≤ d ( d − . Acknowledgments
All authors were partially supported by Hue University’s Project under grant