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On Free Integral Extensions Generated by One Element

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On Free Integral Extensions Generated by One Element On free integral extensions generated by one element IRENA SWANSON, Department of Mathematical sciences, New Mexico State University, Las Cruces, USA. E-mail: [email protected] ORLANDO VILLAMAYOR, Departamento de Matem´aticas, Facultad de Ciencias, C-XV, Universidad Aut´onoma de Madrid, ctra. de Colmenar Viejo, Km. 15, 28049 Madrid, Spain. E-mail: [email protected] Let R be a commutative integral domain with unity, and θ an element of an extension domain satisfying the relation d d−1 d−2 θ = a1θ + a2θ + · · · + ad−1θ + ad; ∼ d d d−i with ai 2 R. We assume throughout that R[θ] = R[X]=(X − i=1 aiX ), where X is an indeterminate over R. Suppose that R is a normal domain with quotient field KP, and K ⊂ L an algebraic extension. Let R be the integral closure of R in L, and fix θ 2 R. There is information on the element θ encoded in the coefficients ai. The first example arises when characterizing if θ belongs to the integral closure of the extended ideal IR, for some ideal I in R. The objective of this paper is to study more precisely what information about θ is encoded in the coefficients ai. i In a first approach, in Section 2, we show that for an ideal I in R, ai 2 I for all i implies that θnR[θ] \ R 2 In for all n, but that the converse fails. Thus contractions of powers of θnR[θ] to R contain some information, but not enough. We turn to a different approach in Sections 3 and 4, where we replace contractions by the trace functions (the image of θnR[θ] in R by the trace function), and it turns out that if θ is separable over K, then the Trace codes more information. The main results in this paper are: a) Propositions 3.6 and 3.8 with conditions that assert that θ belongs to the integral closure of an extended ideal, and b) Propositions 3.12 and 3.14 with conditions that assert that θ belongs to the tight closure of an extended ideal. In all these Propositions we fix an ideal I ⊂ R and consider the extended ideal I:R[θ]. It should be pointed out that normally the condition for θ to belong to the integral closure of I:R[θ], is expressed in terms of a polynomial with coefficients in the ring R[θ]; whereas we will express the same fact but in terms of a polynomial with coefficients in R; furthermore, in terms of the minimal polynomial of θ over R in case R is normal. We also point out that we start with an ideal I in R, and an element θ in R, and we study if θ belongs to integral or tight closure of the extended ideal, but only for the extension R ⊂ R[θ]. This situation is however quite general, at least if I is a parameter ideal. In fact, given a complete local reduced ring (B; M) of dimension d containing a field, and with residue field k, and given a system of parameters fx1; : : : ; xdg, then B is finite over the subring R = k[[x1; : : : ; xd]]. Furthermore an element θ 2 B is in the integral closure (in the tight closure) of the parameter ideal < x1; : : : ; xd > B, if and only if it is so in < x1; : : : ; xd > R[θ]. Throughout the previous argumentation there is a difference between characteristic zero and positive characteristic. The point is that our argu- ments will rely on properties of the subring of symmetric polynomials in a polynomial ring. The relation of symmetric polynomials with our problem will arise and be discussed in the paper. We will show that the properties of θ that we are considering can be expressed in terms of symmetric functions on the roots of the minimal polynomial of θ, and hence as functions on the coefficients ai of the minimal polynomial. If k is a field of characteristic zero and S is a polynomial ring over k, the subring of symmetric polynomials of S can be generated in terms of the trace; however this is not so if k is of positive characteristic. In Section 4 we address the pathological behaviour in positive characteristic, and we give an example in which R is a k-algebra, k a field of positive characteristic, and the k-subalgebra generated by all the T r(θn), as n varies, is not finitely generated. We try to develop our results in maximal generality, in order to distin- guish properties that hold under particular conditions (e.g. on the charac- teristic of R, separability of θ over K, etc.). Our arguments rely on a precise expression of the powers θn of θ in terms of the natural basis f1; θ; θ2; : : : ; θd−1g of R[θ] over R. This is done in Section 1 by using compositions, that is, ordered tuples of positive integers. Similarly, we also develop a product formula for elements of R[θ] in terms of the natural basis. 1 Power and product formula Every element of R[θ] can be written uniquely as an R-linear combination of 1; θ; θ2; : : :, θd−1. In this section we develop formulas for the R-linear combinations for all powers of θ, and for linear combinations of products. Definition 1.1 Let e be a positive integer. A composition of e is an ordered tuple (e1; : : : ; ek) of positive integers such that ei = e. Let E e denote the set of all compositions of e. P For example, E 1 = f(1)g, E 2 = f(2); (1; 1)g, E 3 = f(3); (2; 1); (1; 2); (1; 1; 1)g. We will express θn in terms of these compositions. Without loss of gen- erality we may use the following notation: Notation 1.2 For i > d, set ai = 0. Definition 1.3 Set C0 = 1, and for all positive integers e set Ce = ae1 ae2 · · · aek : (e1;:::X;ek)2E e Remark 1.4 It is easy to see that for all e > 0, Ce = C0ae + C1ae−1 + · · · + Ce−1a1. Proposition 1.5 For all e ≥ 0, d−1 d+e i θ = (C0ad+e−i + C1ad+e−i−1 + C2ad+e−i−2 + · · · + Cead−i) θ : Xi=0 Proof: The proof follows by induction on e. When e = 0, the coefficient of i θ in the expression on the left above is C0ad−i = ad−i, so the proposition holds for the base case by definition. Now let e > 0. Then θd+e = θd+e−1θ d−1 i+1 = (C0ad+e−i−1 + C1ad+e−i−2 + C2ad+e−i−3 + · · · + Ce−1ad−i) θ Xi=0 d−2 i+1 = (C0ad+e−i−1 + C1ad+e−i−2 + C2ad+e−i−3 + · · · + Ce−1ad−i) θ i=0 X d + (C0ae + C1ae−1 + C2ae−2 + · · · + Ce−1a1) θ d−1 i = (C0ad+e−i + C1ad+e−i−1 + C2ad+e−i−2 + · · · + Ce−1ad−i+1) θ Xi=1 d−1 d−1 e i i + Ce ad−iθ = Cjad+e−i−jθ : Xi=0 Xi=0 Xj=0 d+e Recall that ai = 0 if i > d. Thus in the expression for θ in the proposition above, many of the terms Cjad+e−i−j are trivially zero. We similarly determine the product formula: d−1 i d−1 i Let f = i=0 fiθ , g = i=0 giθ be two elements in R[θ]. Write fg d−1 as an R-linear combination of 1; θ; : : : ; θ . (Here, fi = gi = 0 if i < 0 or i ≥ d.) P P 2d−2 d−1 i fg = fkgi−kθ Xi=0 Xk=0 d−1 d−1 2d−2 d−1 i i = fkgi−kθ + fkgi−kθ Xi=0 Xk=0 Xi=d Xk=0 d−1 d−1 d−2 d−1 i d+e = fkgi−kθ + fkgd+e−kθ Xi=0 Xk=0 Xe=0 Xk=0 d−1 d−1 d−2 d−1 d−1 e i i = fkgi−kθ + fkgd+e−k Cjad+e−i−jθ e=0 Xi=0 Xk=0 X Xk=0 Xi=0 Xj=0 d−1 d−1 d−2 e i = f g − + f g − C a − − θ 0 k i k k d+e k j d+e i j1 e=0 Xi=0 Xk=0 X Xj=0 d−1 d−1 @ d−2 e A i = f g − + g − C a − − θ : k 0 i k d+e k j d+e i j1 e=0 Xi=0 Xk=0 X Xj=0 @ A We will use this expression mainly for the cases when fg 2 R. Then the coefficients of θi in the expression above, for i > 0, are 0, and the constant coefficient is d−1 k−1 fk g−k + gd+e−k(C0ad+e + C1ad+e−1 + · · · + Cead) e=0 ! Xk=0 X d−1 k−1 = f0g0 + fk gd+e−kCead: e=0 Xk=0 X 2 Contractions i n In this section we examine implications between ai 2 I for all i, and θ R[θ]\ R 2 In for all n, where I is an ideal of R. In case R is an N-graded ring i with R = R0[R1] and I = R1R, then ai 2 I is equivalent to saying that deg(ai) ≥ i. (The two statements are not equivalent in general.) We examine how under some N-gradings on R, the degrees of the ai affect and are affected by the degrees of the elements of θnR[θ] \ R.
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