6 Ju l 2 00 3 MONOMIAL IDEALS WHOSE POWERS HAVE A LINEAR RESOLUTION

In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. It is known [7] that polymatroidal ideals have linear resolutions and that powers of polymatroidal ideals are again polymatroidal (see [2] and [8]). In particular they have again linear resolutions. In general however, powers of ideals with linear resolution need not to have linear resolutions. The first example of such an ideal was given by Terai. He showed that over a base field of characteristic 6= 2 the Stanley Reisner ideal I = (abd, abf, ace, adc, aef, bde, bcf, bce, cdf, def) of the minimal triangulation of the projective plane has a linear resolution, while I has no linear resolution. The example depends on the characteristic of the base field. If the base field has characteristic 2, then I itself has no linear resolution. Another example, namely I = (def, cef, cdf, cde, bef, bcd, acf, ade) is given by Sturmfels [13]. Again I has a linear resolution, while I has no linear resolution. The example of Sturmfels is interesting because of two reasons: 1. it does not depend on the characteristic of the base field, and 2. it is a linear quotient ideal. Recall that an equigenerated ideal I is said to have linear quotients if there exists an order f1, . . . , fm of the generators of I such that for all i = 1, . . . , m the colon ideals (f1, . . . , fi−1) : fi are generated by linear forms. It is quite easy to see that such an ideal has a linear resolution (independent on the characteristic of the base field). However the example of Sturmfels also shows that powers of a linear quotient ideal need not to be again linear quotient ideals. On the other hand it is known (see [3] and [9]) that the regularity of powers I of a graded ideal I is bounded by a linear function an + b, and is a linear function for large n. For ideals I whose generators are all of degree d one has the bound reg(I) ≤ nd + regx(R(I)), as shown by Römer [12]. Here R(I) is the Rees ring of I which is naturally bigraded, and regx(R(I)) is the x-regularity of R(I). It follows from this formula that each power of I has a linear resolution if regx(R(I)) = 0. In this paper we will show (Theorem 3.2) that if I ⊂ K[x1, . . . , xn] is a monomial ideal with 2-linear resolution, then each power has a linear resolution. Our proof is based on the formula of Römer. In the first section we give a new and very short proof of his result, and remark that if there is a term order such that the initial ideal of the defining ideal P of the Rees ring R(I) is generated by monomials which are linear in the variables x1, . . . , xn, then regx(R(I)) = 0.