Báo cáo toán học: On the Symmetric and Rees Algebras of Some Binomial Ideals

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Báo cáo toán học: " On the Symmetric and Rees Algebras of Some Binomial Ideals"Vietnam Journal of Mathematics 34:1 (2006) 63–70 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 On the Symmetric and Rees Algebras of Some Binomial Ideals Ha Minh Lam1 and Morales Marcel1,2 1 Universit´ de Grenoble I, Institut Fourier, e URA 188, B.P.74, 38402 Saint-Martin D’H`res Cedex, France e 2 IUFM de Lyon, 5 rue Anselme, 69317 Lyon Cedex, France Received April 18, 2005 Revised September 5, 2005Abstract. We give an explicit form of the presentation ideal of the Rees algebra anda primary decomposition of the presentation ideal of the Symmetric algebra for somebinomial ideals generated by four elements, without any assumption on the finitenessand the characteristic of the ground field.IntroductionIn this paper we consider a binomial ideal I in the polynomial ring K[x1 , x2 ,. . . , xn ], minimally generated by four binomials, such that each binomial is adifference of monomials without common factors. Codimension 2 lattice idealsgenerated by four elements are a particular case. We study the Rees algebra andthe Symmetric algebra associated to I. The Rees algebra R(I ) of I is defined to be the graded ring R[It] = k≥0 I k tk .By introducing four independent variables, called T = {T1 , T2 , T3 , T4 }, and con-sidering the ideal J = ker π, where π R[T ] −→ R[It] −→ 0 , Ti −→ fi twe have a presentation R[It] K[x, T ]/J of the Rees algebra. The Symmetricalgebra Sym(I ) of I is Sym(I ) = K[x, T ]/L, where L is the ideal generated bythe first syzygies of I. In this paper, an explicit form of the presentation ideal J will be given64 Ha Minh Lam and Morales Marcelin Theorem 2.1. We obtain also a primary decomposition of the presentationideal of the Symmetric algebra Sym(I ) in Theorem 3.1. All these results areindependent of the characteristic and of the cardinal of K.1. PreliminariesLet fu and fv be two arbitrary binomials in the polynomial ring K[x1 , x2 , . . . , xn ],such that the greatest common divisor (g.c.d. for short) of two terms of eachbinomial is 1. Denote by xp the g.c.d. of the first term of fu and the first termof fv , by xt the g.c.d. of the first term of fu and the second term of fv , by xrthe g.c.d. of the second term of fu and the second term of fv , and by xs theg.c.d. of the second term of fu and the first term of fv . We have fu = α1 xp xt xμ+ − β1 xr xs xμ− (1) fv =α2 xp xs xν+ − β2 xr xt xν−where α1 , α2 , β1 , β2 are non-zero elements in the field K.Remark 1.• The monomials xp , xt , xr , xs are pairwise coprime.• The monomials xμ+ , xν+ , xμ− , xν− are pairwise coprime.• (xt , xμ− ) = (xt , xν+ ) = 1, and (xs , xμ+ ) = (xs , xν− ) = 1, and (xp , xμ− ) =(xp , xν− ) = 1, and (xr , xμ+ ) = (xr , xν+ ) = 1. Consider two new binomials, denoted by fu+v and fu−v , obtained from fuand fv as follows fu+v = α1 α2 x2p xμ+ xν+ − β1 β2 x2r xμ− xν− (2) fu−v = α1 β2 x2t xμ+ xν− − α2 β1 x2s xμ− xν+We denote by I the ideal (fu , fv , fu+v , fu−v ).Example Let L be a lattice in Zn . The lattice ideal IL associated to L is definedas follows IL := (fv := xv+ − xv− | v = v+ − v− ∈ L) ⊂ R := K[x1 , . . . , xn ].If IL is of codimension 2 and is generated by four elements then it is knownthat IL is generated by four binomials of the type fu , fv , fu+v , fu−v as inour case. Moreover, these four binomials are determined by the Hilbert basis{u, v, u + v, u − v } of L.Proposition 1.1. If one of four monomials xp , xt , xr , xs is a unit, then I isof codimension 2, and is either a complete intersection or an almost completeintersection. In both cases, the Rees algebra and the Symmetric algebra areisomorphic.Proof. Assume that one of four monomials xp , xt , xr , xs is a unit. Becausethe role of these four monomials is the same, we can assume that xp = 1. InSymmetric and Rees Algebras of Some Binomial Ideals 65this case, we have fu−v = β2 xν− xt fu − β1 xμ− xs fv , and I becomes the idealgenerated by all the 2 × 2 minors of the matrix ⎛ ⎞ β2 xν− xr −α2 xν+ ⎝ α1 xμ+ −β1 xμ− xr ⎠ . s xt −xHence, we have the following relations β2 xν− xr fu + α1 xμ+ fv − xs fu+v = 0, α2 xν+ fu + β1 xμ− xr fv − xt fu+v = 0.Let us remark that if either xs = 1 or xt = 1 then I is a complete intersectionideal, generated by fu and fv . In this case, it is known that R(I ) = Sym(I ) =K[x, T ]/(fu Tv − fv Tu ). Consider the case where both xs and xt are non units. Set L1 = β2 xν− xr Tu + α1 xμ+ Tv − xs Tu+v , and L2 = α2 xν+ Tu + β1 xμ− xr Tv − tx Tu+v . We have that all these forms are in the presentation ideal J of the Reesring of I. Denote by A the ideal (L1 , L2 ). It is cl ...