High degree anti-integral extensions of Noetherian domains

Abstract

Introduction. Let R be a Noetherian integral domain and R [X] a polynomial ring. Let a be an element of an algebraic field extension L of the quotient field K of R and let π : R [X] -> R [a] be the Λ-algebra homomorphism sending X to a. Let φΛ(X) be the monic minimal polynomial of a over K with deg φΛ(X)=d and write φΛ(X)=X d+ηlX -+ +ηd. Let 7ω:= Π ί,ι(R:R ?,). Foτf(X)^R[X], let C(f(X)) denote the ideal generated by the coefficients of f ( X ) . Let/[Λ]: =/[*] C(φΛ(X)), which is an ideal of R and contains /[*]. The element a is called an anti-integral element of degree d over R if Kerτr= /[rt] φΛ(^C) R [X]. When a is an anti-integral element over Ry R[a] is called an anti-integral extension of R. In the case K(a)=K, an anti-integral elemet a is the same as an anti-integral element (i.e., R=R[a] Γ\R[l/(X\) defied in [5]. The element a is called a super-primitive element of degree d over R if JιΛ^p for all primes p of depth one. For p^ Spec (R), k(p) denotes the residue field Rp/pRp and rank^) R [a] ®R k(p) denotes the dimension as a vector space over k(p). We are interested in characterizing the flatness and the integrality of an anti-integral extension R\cί\ of R. Indeed, among others we obtain the following results: (i) R [a] is flat over R if and only if rank^) R [a] ®R k(p)