LINKAGE EXTENSIONS
NICOLAE MANOLACHE
Given two equidimensional Cohen-Macaulay local rings A1 and A2 of the same dimension, we show that a simultaneous extension ofA1by the dualizing module of A2 and of A2 by the dualizing module of A1, is Gorenstein. This extends a theorem of Fossum [9]. The geometrical analogue of this is also considered. As an example, the pairs of double lines inP3 which are l.a.l. linked are classified. This extends a result of Migliore [18].
AMS 2000 Subject Classification: 14M06, 13C40.
Key words: algebraic variety, algebraic scheme, Cohen-Macaulay ring, Gorenstein ring, locally complete intersection ring, dualizing sheaf, multiple struc- ture.
1. INTRODUCTION
Generalizing a theorem of Reiten [24], Fossum [9] proves that the exten- sion of a Cohen-Macaulay ring by a canonical module is a Gorenstein ring.
This idea is considered in a geometrical frame by Ferrand [8], mainly for curves in a threefold. Both these considerations can be interpreted in the frame of
“algebraic linkage” (cf. [23]). 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 exten- sion”. In a natural way one can extend the notion to the case of schemes.
In general, given two (embedded) schemes, there is no (embedded) link- age extension of them. In order to illustrate the “rarity” of linkage extensions, we classify the pairs of double lines inP3 which are locally algebraically linked.
2. LINKED EXTENSION 2.1. The algebraic case
All the rings considered here are commutative and noetherian. IfA is a local ring, we denote bymA its maximal ideal.
REV. ROUMAINE MATH. PURES APPL.,52(2007),4, 397–403
Definition 2.1. Let A1,A2 be two rings, M1 an A1-module, M2 an A2- module. We calldouble extension of A1 by M2 and A2 by M1 a couple
0→M2→i2 B →p1 A1 →0, 0→M1 →i1 B →p2 A2→0
of exact sequences, where B is a commutative ring, the maps p1, p2 are ring homomorphisms and for allb∈B,x1∈M1,x2 ∈M2 we have
bi1(x1) =i1(p1(b)x1), bi2(x2) =i2(p2(b)x2).
By abuse of language we also call 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 ringB such that it can be inserted in extensions as above.
Example 2.2. For each surjective mapB →B/aof rings we realize canon- icalyB 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. WhenA1,A2 in the above definition are equidimensional Cohen-Macaulay rings of the same dimension, and M1, M2 are respectively the dualizing modulesω1,ω2, a double extension ofA1 by ω2 and ofA2 byω1
is calledlinkage extension of A1 and A2. This means a couple 0→ω2→i2 B →p1 A1 →0, 0→ω1→i1 B →p2 A2 →0
of exact sequences, where B is a commutative ring, the maps p1, p2 are ring homomorphisms and for allb∈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 ringB like above a linkage extension of A1 and A2.
Theorem2.4. 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.
Proof. The ring B is Cohen-Macaulay of the dimension of A1 and A2. We shall make an induction on this dimension. First, two auxiliary results.
Lemma 2.5. In the situation from Definition 2.3, an element b∈mB is B-regular 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 Cohen-Macaulay 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(socleωA1) =i2(socleωA2) = socleB.
Proof. By symmetry, it is enough to show i1(socleωA1) = socleB. For this, let b ∈ socleB. Then bmB = 0 and so bi1(ωA1) = 0. By definition this gives i1(p1(b)ωA1) = 0 and so p1(b)ωA1 = 0. As AnnA1ωA1 = 0, it follows p1(b) = 0. This proves the existence of an element b2 ∈ ωA2 such that b = i2(b2). But 0 = bmB = i2(b2)mB = i2(b2mB), so b2mB = 0, i.e., b2 ∈ socleωA2. This shows b ∈ i2(socleωA2). Thus, we proved socleB ⊂ i2(socleωA2). The other inclusion is evident.
We come back to the proof of Theorem 2.4. If dimB ≥1 letb∈B be a regular element. Thenp1(b) is regular inA1andωA1 andp2(b) is regular inA2
andωA2 and one shows easily thatB/bB is a linked extension ofA1/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, asωAi ∼=E(Ai/mi) (the injective envelope of the residue field ofAi) and the socle of ωAi is Ai/mi, it follows socleB = Ai/mi i.e., the socle of B is simple. This proves thatB is Gorenstein.
Remark 2.7. WhenA1=A2, we obtain a theorem of Fossum [9] accord- ing to which any extension of a local Cohen-macaulay ringAby its dualizing module is Gorenstein. We shall call such a ringa (Fossum) doubling of A. In fact the proof given here is “a splitting” of the original proof (cf. [9]). In a geometric frame the similar construction is known asFerrand’s doubling. This terminology is motivated by the fact that the multiplicity of a doubling of a Cohen-Macaulay ring is twice 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 +x1x2, x22) is a linkage extension of A1 := R/a1 and A2 := R/a2. The interest of this example lies in the fact thatA1,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 also realized as a linkage extension ofAand a triple structure onA.
2.2. The geometric case
Definition 2.9. If X1, X2 are two algebraic schemes, we call double ex- tensionof them a couple
0→ F2 i2
→ OY → Op1 X1 →0, 0→ F1 i1
→ OY → Op2 X2 →0
of exact sequences, whereFi areOXi-modules,i= 1,2, andY 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 calllinkage extension ofX1 andX2 a double extension
0→ω2⊗L2→ Oi2 Y p1
→ OX1 →0, (1)
0→ ω1⊗L1→ Oi1 Y p2
→ OX2 →0 (2)
of the shape, whereLiare invertibleOXi-modules andωiare the corresponding dualizing sheaves,i= 1,2. By abuse of language we say also thatY is alinkage extension ofX1,X2.
Theorem 2.10. If X1, X2 are equidimensional locally Cohen-Macaulay schemes of the same dimension, any linkage extension of them is locally Goren- stein.
Proof. The question is local, so Theorem 2.4 applies.
Lemma 2.11. If X is a linkage extension of two equidimensional locally Cohen-Macaulay schemes X1, X2 of the same dimension, then, with the no- tation from Definition2.9, we have
L1=ω−1Y |X1, L2=ω−1Y |X2.
Proof. Applying the dualizing functorHom(?, ωY) to the exact sequence (1) and then tensoring withω−1Y yield
0→ω1⊗ωY−1 → OY → OX2 ⊗L−12 ⊗ωY−1→0.
HereOX2⊗L−12 ⊗ωY−1is the structural sheaf of a closed subscheme ofY which coincides locally withX2, so it is OX2. Thenω1⊗ω−1Y =ω1⊗L1. This shows L1∼=ω−1Y |X1. Analogously,L2∼=ω−1Y |X2.
Remark 2.12. If, givenX1,X2, there exists a linkage extension Y ofX1, X2, we also say that X1 and X2 are locally Gorenstein linked. When Y is locally complete intersection, we say thatX1 and X2 are locally algebraically linked, l.a.l. for short, (cf. [13], where this terminology was introduced, in- spired by the notion ofalgebraic linkage of [23]).
If both X1 and X2 are embedded in a (let say smooth) scheme P and we require that the linkage extensionY be also a closed subscheme of P, we say thatY is an embedded(in P) linkage extension ofX1 and X2. In general, givenX1 and X2 inP, there is no linkage extension (in P) of them.
The aim of the next theorem is to classify the pairs of double lines inP3 which are locally algebraically linked.
Theorem2.13. If two double linesY1,Y2inP3with 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 ofP3) they are defined by ideals of the form
IY1 = (ax+by, x2, xy, y2), IY2 = (cx+dz, x2, xz, z2),
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)= 0, d(0 : 1)= 0,or
(b) b(0 : 1) = 0, d(0 : 1) = 0 and a(0 : 1)∂b∂z(0 : 1) =c(0 : 1)∂d∂y(0 : 1);
(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, y2), IY2 = (ax−by, x2, xy, y2),
where a(z, u), b(z, u) are homogeneous forms in z, u of the same degree r. Proof. We have to show that if X1 and X2 meet at a point, then (ii) is fulfilled while ifX1 =X2, then (iii) is fulfilled.
IfX1,X2 have a point in common, one may suppose that the equations ofX1 and X2 arex=y = 0, respectivelyx=z= 0, in suitable homogeneous coordinates (x : y :z :u). Then double structures Y1,Y2 on them are given by ideals as those in (ii). We have to determine whenY1∪Y2 is a complete intersection in X1 ∩X2 = (0 : 0 : 0 : 1). In the affine space u = 0 with coordinatesξ = xu,η= uy,ζ= zu, one has
IY1 = (αξ+βη, ξ2, ξη, η2), IY2 = (γξ+δζ, ξ2, ξζ, ζ2), whereα,β are functions of ζ and γ,δ are functions ofη.
A direct computation shows that if only one of the β(0), δ(0) is 0, then Y1∪Y2 is not a complete intersection inY1∩Y2.
If β(0) = 0, δ(0) = 0 and with new parameters Y = αξ +βη and Z =γξ+δζ, the ideals are local: IY1 = (Y, ξ2), IY2 = (Z, ξ2) and IY1∪Y2 = (Y Z, ξ2). This yields case (a).
Considerβ(0) = 0,δ(0) = 0, i.e. β =β1ζ,δ =δ1η. Then α(0)= 0 and γ(0)= 0. The ideal of Y1∪Y2 is then the intersection
(α(0)ξ+β1(0)ζη+Lξζ+M ζ2η, ξ2, ξη, η2) ∩
∩(γ(0)ξ+δ1(0)ηζ+N ξη+P η2ζ, ξ2, ξζ, ζ2), i.e., it is
((α(0)ξ+β1(0)ζη, ξ2, ξη, η2)∩(γ(0)ξ+δ1(0)ηζ+N ξη+P η2ζ, ξ2, ξζ, ζ2).
This ideal defines a complete intersection iff there exists λ = 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. So, there exists a multiplicity 4 structureY on X which is a linked extension of Y1 and Y2. Consider first the case where Y is a quasiprimitive structure in the sense of B˘anic˘a and Forster (cf. [2] or [14]). Then Y is a doubling of a doublingY ofX (cf. [14], Lemma 2.10). The curveY is obtained canonically from Y, it being a member of the filtrations introduced in [2], [13] and [14], 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 both Y1
and Y2 are 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 [1] or [2] it is a globally complete intersection and, in suitable coordinates, its ideal is IY = (x2, y2). In these coordinates, with suitable 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, y2). Then the ideal ofY2should beIY :I1= (ax−by, x2, xy, y2).
Acknowledgements. The author was partially supported by Contract CEx05-D11- 11/2005. Thanks are also due to the Institute of Mathematics of the University of Oldenburg, especially to Professor U. Vetter, for warm hospitality in December 2004, when last touches to this paper were done.
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Received 18 December 2006 Romanian Academy
“Simion Stoilow” Institute of Mathematics P.O. Box 1-764
014700 Bucharest, Romania [email protected]
