arXiv:math/0501036v1 [math.AG] 3 Jan 2005
Linkage Extensions Nicolae Manolache Abstract Given two equidimensional Cohen-Macaulay local rings of the same dimension, A1 and A2 , we show that a simultaneous extension of A1 by the dualizing module of A2 and of A2 by the dualizing module of A1 is Gorenstein. This extends a theorem of Fossum (cf. [Fo]). The geometrical analogue of this is also considered. As an example, the pairs of double lines in P3 which are l.a.l linked are classified. This extends a result of Migliore (cf. [Mi]). Key words: Algebraic variety, algebraic scheme, Cohen-Macaulay ring, Gorenstein ring, locally complete intersection ring, dualizing sheaf, multiple structure.
1
Introduction
In [Fo], generalizing a theorem of Reiten (cf. [R]), Fossum proves that the extension of a Cohen-Macaulay ring by a canonical module is a Gorenstein ring. This idea is considered in a geometrical frame by Ferrand (cf. [Fe]), mainly for curves in a threefold. Both these considerations can be interpreted in the frame of ”algebraic linkage” (cf. [PS]). In this paper we show that, in fact, if a ring extends two Cohen-Macaulay rings, each by a dualizing module of the other, then this extension is Gorenstein. We call such an extension ”linkage extension”. In a natural way one can extend the notion to the case of schemes. In general, given two (embedded) schemes, there is no (embedded) linkage extension of them. In order to illustrate the ”rarity” of linkage extensions, we classify the pairs of double lines in P3 which are locally algebraically linked.
2 2.1
Linked extension The algebraic case
All the rings considered here are commutative and noetherian. If A is a local ring, we denote by mA its maximal ideal. Definition 2.1 Let A1 , A2 be two rings, M1 an A1 -module, M2 an A2 module. We call double extension of A1 by M2 and A2 by M1 a couple of exact sequences: p1 i2 B → A1 → 0 , 0 → M2 → 1
p2
i
1 B → A2 → 0 , 0 → M1 →
where B is a commutative ring, the maps p1 , p2 are ring homomorphisms and, for all b ∈ B, x1 ∈ M1 , x2 ∈ M2 we have: bi1 (x1 ) = i1 (p1 (b)x1 ) bi2 (x2 ) = i2 (p2 (b)x2 ) By abuse of language we call also double extension of A1 by M2 and A2 by M1 a ring B which can be inserted in a double extension of A1 by M2 and A2 by M1 as above. Also, when only A1 and A2 are given, we call double extension of A1 and A2 any ring B with the property that it can be inserted in extensions as above. Example 2.2 For each surjective map of rings B → B/a we realize canonicaly B as a double extension: 0 → a → B → B/a → 0 0 → 0 : a → B → B/(0 : a) → 0 We shall consider a special case of this notion: Definition 2.3 When A1 , A2 in the above definition are equidimensional Cohen-Macaulay rings of the same dimension, and M1 , M2 are respectively the dualizing modules ω1 , ω2 , a double extension of A1 by ω2 and of A2 by ω1 is called linkage extension of A1 and A2 . This means a couple of exact sequences: i
p1
i
p2
2 B → A1 → 0 , 0 → ω2 → 1 B → A2 → 0 , 0 → ω1 →
where B is a commutative ring, the maps p1 , p2 are ring homomorphisms and, for all b ∈ B, x1 ∈ M1 , x2 ∈ M2 we have: bi1 (x1 ) = i1 (p1 (b)x1 ) bi2 (x2 ) = i2 (p2 (b)x2 ) By abuse of language we call also a ring B like above a linkage extension of A1 and A2 Theorem 2.4 Let A1 , A2 be two equidimensional local Cohen-Macaulay local rings, of the same dimension. In a linkage extension of A1 and A2 as above, the ring B is Gorenstein. 2
Proof. The ring B is Cohen-Macaulay of the dimension of A1 and A2 . We shall make an induction on this dimension. First two lemmas: Lemma 2.5 In the situation from Definition 2.3 an element b ∈ mB is Bregular if and only if p1 (b) is regular in A1 and p2 (b) is regular in A2 . Proof. Everything follows from the observation that an element in a CohenMacaulay ring is regular iff it is regular in the dualizing module. Lemma 2.6 If B is a linked extension of two artinian local rings A1 , A2 , then i1 (socle ωA1 ) = i2 (socle ωA2 ) = socle B . Proof. By symmetry, it is enough to show i1 (socle ωA1 ) = socle B. For this, let b ∈ socle B. Then bmB = 0 and so bi1 (ωA1 ) = 0. By definition this gives i1 (p1 (b)ωA1 ) = 0 and so p1 (b)ωA1 = 0. As AnnA1 ωA1 = 0 it follows p1 (b) = 0. This proves the existence of an element b2 ∈ ωA2 so that b = i2 (b2 ). But 0 = bmB = i2 (b2 )mB = i2 (b2 mB ) and so b2 mB = 0, i.e. b2 ∈ socle ωA2 . This shows b ∈ i2 (socle ωA2 ). Thus we proved socle B ⊂ i2 (socle ωA2 ). The other inclusion is evident. We come back to the proof of the Theorem 2.4. If dim B ≥ 1 let b ∈ B be a regular element. Then p1 (b) is regular in A1 and ωA1 and p2 (b) is regular in A2 and ωA2 and one shows easily that B/bB is a linked extension of A1 /p1 (b)A1 and A2 /p2 (b)A2 . As B is Gorenstein iff B/bB is Gorenstein, by repeating the above argument one reduces the question to the artinian case. In this situation, as ωAi ∼ = E(Ai /mi ), (the injective envelope of the residue field of Ai ) and the socle of ωAi is Ai /mi , it follows socle B = Ai /mi i.e. the socle of B is simple. This proves that B is Gorenstein. Remark 2.7 When A1 = A2 , we obtain the Theorem of Fossum (cf. [Fo]) that any extension of a local Cohen-macaulay ring A by its dualizing module is Gorenstein. We shall call such a ring a (Fossum) doubling of A. In fact the proof given here is ”a splitting” of the original proof (cf. [Fo]). In a geometric frame the similar construction is known as Ferrand’s doubling. This terminology is motivated by the fact that the multiplicity of a doubling of a Cohen-Macaulay ring is double the multiplicity of that ring. Example 2.8 Let R = k[x1 , . . . , xn ] with n ≥ 2 and let a1 , a2 be the ideals a1 = (x1 , x22 ), a2 = (x1 + x2 , x22 ). Then B := R/(x21 + x1 x2 , x22 ) is a linkage extension of A1 := R/a1 and A2 := R/a2 . The interest of this example lies in the fact that A1 , A2 are two ”Fossum doublings” of (A1 )red = (A2 )red = k[x3 , . . . xn ] =: A and B is a multiplicity 4 structure on A which is not a doubling of A1 or A2 . In fact B can be realized as a doubling of A3 = R/(x21 , y). It can be realized also as a linkage extension of A and a triple structure on A.
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2.2
The Geometric Case
Definition 2.9 If X1 , X2 are two algebraic schemes, we call double extension of them a couple of exact sequences: i
p1
i
p2
2 OY → OX1 → 0 , 0 → F2 → 1 OY → OX2 → 0 , 0 → F1 →
where Fi are OXi -modules and Y is an algebraic scheme, such that locally one has a double extension in the sense of 2.1 If X1 , X2 are equidimensional locally Cohen-Macaulay schemes, of the same dimension, we call linkage extension of X1 and X2 a double extension of the shape: i
p1
i1
p2
2 OY → OX1 → 0 , 0 → ω2 ⊗ L2 →
0 → ω1 ⊗ L1 → OY → OX2 → 0 ,
(1) (2)
where , Li are invertible OXi -modules and ωi are the corresponding dualizing sheaves. By abuse of language we say also that Y is a linkage extension of X1 , X2 . Theorem 2.10 If X1 , X2 are equidimensional locally Cohen-Macaulay schemes, of the same dimension, any linkage extension of them is locally Gorenstein. Proof. The question is local, so 2.4 applies.
Lemma 2.11 If X is a linkage extension of two equidimensional locally CohenMacaulay schemes X1 , X2 of the same dimension, then, with the notation from the definition 2.9: L1 = ωY−1 |X1 , L2 = ωY−1 |X2 . Proof. Applying the dualizing functor Hom(?, ωY ) to the exact sequence (1) and then tensoring with ωY−1 one gets: −1 0 → ω1 ⊗ ωY−1 → OY → OX2 ⊗ L−1 2 ⊗ ωY → 0 −1 Here OX2 ⊗ L−1 2 ⊗ ωY is the structural sheaf of a closed subscheme of Y which coincides locally with X2 , so it is OX2 . Then ω1 ⊗ ωY−1 = ω1 ⊗ L1 . This shows L1 ∼ = ωY−1 |X2 . = ωY−1 |X1 . Analogously L2 ∼
Remark 2.12 If, given X1 , X2 , there exists a Y which is a linkage extension of X1 , X2 , we say also that X1 and X2 are locally Gorenstein linked. When Y is locally complete intersection we say that X1 and X2 are locally algebraically linked, l.a.l. for short, (cf. [M2], where this terminology was introduced, inspired by the notion of algebraic linkage of [PS]).
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If X1 and X2 are both embedded in a (let say smooth) scheme P and we require that the linkage extension Y to be also closed subscheme of P , we say that Y is an embedded (in P ) linkage extension of X1 and X2 . In general, given X1 and X2 in P there is no linkage extension (in P ) of them. The aim of the next theorem is to classify the pairs of double lines in P3 which are locally algebraically linked. Theorem 2.13 If two double lines Y1 , Y2 in P3 with supports respectively X1 , X2 , are l.a.l. then they are in one of the following situation: (i) Y1 , Y2 are disjoint (ii) in convenient homogeneous coordinates (x : y : z : u) (i.e. after an automorphism of P3 ) they are defined by ideals of the form: IY1 = (ax + by, x2 , xy, y 2 ) IY2 = (cx + dz, x2 , xz, z 2 ) , where a(z, u), b(z, u) are homogeneous forms in z, u of the same degree r1 , without common zeros on X1 , c(y, u), d(y, u) are homogeneous forms in y, u of the same degree r2 , without common zeros on X2 , such that: (a) b(0 : 1) 6= 0, d(0 : 1) 6= 0, or (b) b(0 : 1) = 0, d(0 : 1) = 0 and a(0 : 1)
∂b ∂d (0 : 1) = c(0 : 1) (0 : 1) ∂z ∂y
(iii) if X1 = X2 then, either (a) Y1 = Y2 , or (b) in convenient homogeneous coordinates (x : y : z : u) they are defined by ideals of the form: IY1 = (ax + by, x2 , xy, y 2 ) IY2 = (ax − by, x2 , xy, y 2 ) , where a(z, u), b(z, u) are homogeneous forms in z, u of the same degree r. Proof. We have to show that, when X1 and X2 meet in a point, (ii) is fulfilled and that, when X1 = X2 , (iii) is fulfilled. If X1 , X2 have a point in common, one may suppose that the equations of X1 and X2 are x = y = 0, respectively x = z = 0 in convenient homogeneous coordinates (x : y : z : u). Then double structures Y1 , Y2 on them are given by ideals like those in (ii). We have to determine when Y1 ∪ Y2 is a complete
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intersection in X1 ∩ X2 = (0 : 0 : 0 : 1). In the affine space u 6= 0, with x y z coordinates ξ = , η = , ζ = , one has : u u u IY1 = (αξ + βη, ξ 2 , ξη, η 2 ) IY2 = (γξ + δζ, ξ 2 , ξζ, ζ 2 ) , where α, β are functions in ζ and γ, δ are functions in η. A direct computation shows that, if only one of the β(0), δ(0) is 0, then Y1 ∪ Y2 is not a complete intersection in Y1 ∩ Y2 . If β(0) 6= 0, δ(0) 6= 0 and with new parameters Y = αξ +βη and Z = γξ +δζ, the ideals are locally: IY1 = (Y, ξ 2 ), IY2 = (Z, ξ 2 ) and IY1 ∪Y2 = (Y Z, ξ 2 ). This hives the case (a). Consider β(0) = 0, δ(0) = 0, i.e. β = β1 ζ, δ = δ1 η. Then α(0) 6= 0 and γ(0) 6= 0. The ideal of Y1 ∪ Y2 is then the intersection : (α(0)ξ + β1 (0)ζη + Lξζ + M ζ 2 η, ξ 2 , ξη, η 2 ) ∩ (γ(0)ξ + δ1 (0)ηζ + N ξη + P η 2 ζ, ξ 2 , ξζ, ζ 2 ) , i.e. it is ((α(0)ξ + β1 (0)ζη, ξ 2 , ξη, η 2 ) ∩ (γ(0)ξ + δ1 (0)ηζ + N ξη + P η 2 ζ, ξ 2 , ξζ, ζ 2 ) . This ideal defines a complete intersection iff there exists a λ 6= 0 such that α(0)ξ + β1 (0)ζη = λ(γ(0)ξ + δ1 (0)ηζ). This condition translates to (b). Consider now the case when X1 = X2 =: X. If Y1 and Y2 are l.a.l., then there exists a multiplicity 4 structure Y on X which is a linked extension of Y1 and Y2 . Consider first the case Y is a quasiprimitive structure in the sense of B˘anic˘a and Forster (cf. [BF2] or [M3]). Then Y is a doubling of a doubling Y ′ of X (cf. [M3], Lemma 2.10). The curve Y ′ is obtained canonically from Y , being a member of the filtrations introduced in [BF2], [M2] and [M3], which coincide for quasiprimitive structures. In the B˘anic˘a- Forster filtration Y ′ is obtained throwing away the embedded points of Y ∪X (2) . As Y1 and Y2 are both contained in X (2) , they should be contained in Y ′ , i.e. one has Y1 = Y2 = Y ′ . If Y is not a quasiprimitive structure, then according to [BF1] or [BF2] it is globally complete intersection, and, in convenient coordinates, its ideal is IY = (x2 , y 2 ). In these coordinates, with convenient forms a(z, u), b(z, u) of the same degree, without common zeros along X, the ideal of X1 is I1 = (ax+by, x2 , xy, y 2 ). Then the ideal of Y2 should be IY : I1 = (ax−by, x2 , xy, y 2 ). Acknowledgement. The author was partially supported by Contract CERES 3-28, 2003-2005 and by Contract CERES 2004-2006. Thanks are due also to the Institute of Mathematics of the University of Oldenburg, especially to Prof. U. Vetter, for the warm hospitality in decmber 2004, when last touches to this paper were done.
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References [BF1]
˘ nica ˘ , O. Forster C. Ba (1981)
[BF2]
˘ nica ˘ , O. Forster : Multiplicity structures on Space Curves , ConC. Ba temporary mathematics , 58 (1986)
[BM]
˘ nica ˘ , N. Manolache : Rank 2 Stable vector bundles on P3 (C) , C. Ba Math. Z. , 190 (1985) , p. 315-339
[BM’]
˘ nica ˘ , N. Manolache : Remarks on rank 2 stable vector bundles on C. Ba P3 with Chern classes c1 = −1, c2 = 4 , Preprint INCREST , 104 (1981)
[BM”]
˘ nica ˘ , N. Manolache : Moduli space M (−1, 4): Minimal spectrum , C. Ba Preprint INCREST , 19 (1983)
[Ba]
H. Bass : On the ubiquity of Gorenstein rings , Math. Z. , 82 (1963) , p. 8-28
[BE]
D. Bayer, D. Eisenbud : Ribbons and their canonical embeddings , Trans. Amer. Math. Soc. , 347 (1995) , p. 757-765
[Fe]
D. Ferrand : Courbes Gauches et Fibr´es de Rang 2 , C.R. Acad. Sci. Paris , 281 (1975) , p. 345-347
[Fo]
R. Fossum : Commutative Extensions by Canonical Modules are Gorenstein Rings , Proc. of the Amer. Math. soc. , 40 (1973) , p. 395-400
[H1]
R. Hartshorne : Stable Vector Bundles of Rank 2 on P3 , Math. Ann. , 238 (1978) , p. 229-280
[H2]
R. Hartshorne : Stable reflexive Sheaves of Rank 2 on P3 , Math. Ann. , 254 (1978) , p. 121-176
[M1]
N. Manolache : Rank 2 Stable Vector Bundles on P3 with Chern classes c1 = −1, c2 = 2 , Rev. Roumaine Math. pures et Appl. , 26 (1981) , p. 1203-1209
[M2]
N. Manolache : Cohen-Macaulay Nilpotent Structures , Rev. Roumaine Math. pures et Appl. , 31 (1986) , p. 563-575
[M3]
N. Manolache : Codimension Two Linear Varieties with Nilpotent Structures , Math. Z. , 210 (1992) , p. 573-580
[M4]
N. Manolache : Curves Y of Degree 6 with ωY = OY (−1) on a Smooth Quadric Q3 , Annali di Matematica pura ed applicata (IV) , 167 (1994) , p. 226-241
[M5]
N. Manolache : Multiple Structures on Smooth Support , Math. Nachr. , 167 (1994) , p. 157-202
[M6]
N. Manolache : Double Rational Normal Curves with Linear Syzygies , Manuscripta Mathematica , 104, No. 4 (2001) , p. 503-517
: Sur les Structures Multiples (manuscript)
7
[Mi]
J. Migliore : On linking double lines , Trans. Am. Math. soc., , 294 (1986) , p. 177-185
[NNS]
U. Nagel, R. Notari, M. L. Spreafico : The Hilbert Scheme of Degree Two Curves and Certain Ropes , arXiv:math.AG/0311537
[N]
S. Nollet : The Hilbert Schemes of Degree Three Curves , Ann. Scient. E.N.S., 4e S´e rie , 30 (1997) , p. 367-384
[NS]
S. Nollet, E. Schlesinger : Hilbert Schemes of Degree Four Curves , arXiv:math.AG/0112167
[OSz]
G. Ottaviani, M. Szurek : On Moduli of Stable 2-Bundles with Small Chern Classes on Q3 , Annali di Matematica pura ed applicata (IV), , 167 (1994) , p. 191-241
[PS]
C. Peskine, L. Szpiro : Liaison des Vari´etes Alg´ebriques , Invent. Math. , 26 (1977) , p. 271-302
[R]
I. Reiten : The Converse to a Theorem of Sharp on Gorenstein Modules , Proc. Amer. math. Soc. , 32 (1972) , p. 417-420
[Sz]
L. Szpiro : Equations Defining Space Curves , Tata institute Lect. Notes (notes by N. Mohan Kumar, Bombay (1979)
[V1]
J. E. Vatne : Towards a Classification of Multiple Structures , Ph.D. thesis, University of Bergen (2001)
[V2]
J. E. Vatne : Multiple Structures , arXiv:math.AG/0210042
[V3]
J. E. Vatne : Monomial Multiple Structures , arXiv:math.AG/0210101
Nicolae Manolache Institute of Mathematics ”Simion Stoilow” of the Romanian Academy P.O.Box 1-764 Bucharest, RO-014700 e-mail:
[email protected]
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