SPECIALIZATION OF GRADED MODULES

Dam Van Nhi
2002 Proceedings of the Edinburgh Mathematical Society  
The paper shows that specializations of finitely generated graded modules are also graded and that many important invariants of graded modules and ideals are preserved by specializations. Dam Van Nhi jective varieties. There we will give a simple proof for the global Bertini Theorem of Flenner [5] . Throughout this paper we assume that all modules are finitely generated. Definition and basic properties Let k be an infinite field of arbitrary characteristic. Denote by K an extension field of k.
more » ... et u = (u 1 , . . . , u m ) be a family of indeterminates and α = (α 1 , . . . , α m ) a family of elements of K. We denote the polynomial rings in n+1 variables x 0 , . . . , x n over k(u) and k(α) by R = k(u) [x] and by R α = k(α) [x], respectively. Let m and m α be the maximal graded ideals of R and R α , respectively. We shall say that a property holds for almost all α if it holds for all points of a Zariski-open non-empty subset of K m . For convenience we shall often omit the phrase 'for almost all α' in the proofs of the results of this paper. Following [20] we define the specialization of I with respect to the substitution u → α as the ideal I α of R α generated by elements of the set {f (α, This definition is slightly different than that considered by Krull and Seidenberg, who choose α ∈ k m . However, if some property holds for almost all α ∈ K m in the sense of Krull and Seidenberg, then it holds for the extensions of I α in the polynomial ring K [x] for almost all α ∈ K m . Since k(α)[x] → K[x] is a flat extension, we can often deduce that this property also holds for almost all I α in our sense. Example 2.
doi:10.1017/s0013091500000602 fatcat:bbv7kwoxz5h4tjepv6fcig3gc4