NORMAL TRANSVERSALITY AND UNIFORM BOUNDS

Abstract

Let A be a commutative ring. A graded A-algebra U = [oplus ]_{n[ges ]0} is a standard A-algebra if U₀ = A and U = A[U₁] is generated as an A-algebra by the elements of U₁. A graded U-module F = [oplus ]_{n[ges ]0}F_n is a standard U-module if F is generated as a U-module by the elements of F₀, that is, F_n = U_nF₀ for all n [ges ] 0. In particular, F_n = U₁F_n−1 for all n [ges ] 1. Given I, J, two ideals of A, we consider the following standard algebras: the Rees algebra of I, [Rscr ](I) = [oplus ]_{n[ges ]0}Iⁿtⁿ = A[It] ⊂ A[t], and the multi-Rees algebra of I and J, [Rscr ](I, J) = [oplus ]_{n[ges ]0}([oplus ]_p+q=nI^pJ^qu^pv^q) = A[Iu, Jv] ⊂ A[u, v]. Consider the associated graded ring of I, [Gscr ](I) = [Rscr ](I) [otimes ] A/I = [oplus ]_{n[ges ]0}Iⁿ/Iⁿ⁺¹, and the multi-associated graded ring of I and J, [Gscr ](I, J) = [Rscr ](I, J) [otimes ] A/(I+J) = [oplus ]_{n[ges ]0}([oplus ]_{p+q= n}I^pJ^q/(I+J)I^pJ^q). We can always consider the tensor product of two standard A-algebras U = [oplus ]_{p[ges ]0}U_p and V = [oplus ]_{q[ges ]0}V_q as a standard A-algebra with the natural grading U [otimes ] V = [oplus ]_{n[ges ]0}([oplus ]_p+q=nU_p [otimes ] V_q). If M is an A-module, we have the standard modules: the Rees module of I with respect to M, [Rscr ](I; M) = [oplus ]_{n[ges ]0}IⁿMtⁿ = M[It] ⊂ M[t] (a standard [Rscr ](I)-module), and the multi-Rees module of I and J with respect to M, [Rscr ](I, J; M) = [oplus ]_{n[ges ]0}([oplus ]_p+q=nI^pJ^qMu^pv^q) = M[Iu, Jv] ⊂ M[u, v] (a standard [Rscr ](I, J)-module). Consider the associated graded module of M with respect to I, [Gscr ](I; M) = [Rscr ](I; M) [otimes ] A/I = [oplus ]_{n[ges ]0}IⁿM/Iⁿ⁺¹M (a standard [Gscr ](I)-module), and the multi-associated graded module of M with respect to I and J, [Gscr ](I, J; M) = [Rscr ](I, J; M) [otimes ] A/(I+J) = [oplus ]_{n[ges ]0}([oplus ]_{p+q= n}I^pJ^qM/ (I+J)I^pJ^qM) (a standard [Gscr ](I, J)-module). If U, V are two standard A-algebras, F is a standard U-module and G is a standard V-module, then F [otimes ] G = [oplus ]_{n[ges ]0}([oplus ]_{p+q= n}F_p [otimes ] G_q) is a standard U [otimes ] V-module.

Denote by π[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I, J; M) and σ[ratio ][Rscr ](I, J; M) → [Rscr ](I+J; M) the natural surjective graded morphisms of standard [Rscr ](I) [otimes ] [Rscr ](J)-modules. Let ϕ[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I+J; M) be σ∘π. Denote by &πmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I, J; M) and &σmacr;[ratio ][Gscr ](I, J; M) → [Gscr ](I+J; M) the tensor product of π and σ by A/(I+J); these are two natural surjective graded morphisms of standard [Gscr ](I) [otimes ] [Gscr ](J)-modules. Let &ϕmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I+J; M) be &σmacr;∘&πmacr;. The first purpose of this paper is to prove the following theorem.