Normal Transversality and Uniform Bounds |
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Authors: | Planas-Vilanova Francesc |
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Affiliation: | Departament de Matemàtica Aplicada 1, ETSEIB, Universitat Politècnica de Catalunya Diagonal 647, E-08028 Barcelona, Spain, planas{at}ma1.upc.es |
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Abstract: | Let A be a commutative ring. A graded A-algebra U = n0 Un isa standard A-algebra if U0 = A and U = A[U1] is generated asan A-algebra by the elements of U1. A graded U-module F = n0Fnis a standard U-module if F is generated as a U-module by theelements of F0, that is, Fn = UnF0 for all n 0. In particular,Fn = U1Fn1 for all n 1. Given I, J, two ideals of A,we consider the following standard algebras: the Rees algebraof I, R(I) = n0Intn = A[It] A[t], and the multi-Rees algebraof I and J, R(I, J) = n0(p+q=nIpJqupvq) = A[Iu, Jv] A[u, v].Consider the associated graded ring of I, G(I) = R(I) A/I =n0In/In+1, and the multi-associated graded ring of I and J,G(I, J) = R(I, J) A/(I+J) = n0(p+q=nIpJq/(I+J)IpJq). We canalways consider the tensor product of two standard A-algebrasU = p0Up and V = q0Vq as a standard A-algebra with the naturalgrading U V = n0(p+q=nUp Vq). If M is an A-module, we havethe standard modules: the Rees module of I with respect to M,R(I; M) = n0InMtn = M[It] M[t] (a standard R(I)-module), andthe multi-Rees module of I and J with respect to M, R(I, J;M) = n0(p+q=nIpJqMupvq) = M[Iu, Jv] M[u, v] (a standard R(I,J)-module). Consider the associated graded module of M withrespect to I, G(I; M) = R(I; M) A/I = n0InM/In+1M (a standardG(I)-module), and the multi-associated graded module of M withrespect to I and J, G(I, J; M) = R(I, J; M) A/(I+J) = n0(p+q=nIpJqM/(I+J)IpJqM)(a standard G(I, J)-module). If U, V are two standard A-algebras,F is a standard U-module and G is a standard V-module, thenF G = n0(p+q=nFp Gq) is a standard U V-module. Denote by :R(I) R(J; M) R(I, J; M) and :R(I, J; M) R(I+J;M) the natural surjective graded morphisms of standard RI) R(J)-modules. Let :R(I) R(J; M) R(I+J; M) be . Denote by :G(I) G(J; M) G(I, J; M) and :G(I, J; M) G(I+J; M) the tensor productof and by A/(I+J); these are two natural surjective gradedmorphisms of standard G(I) G(J)-modules. Let :G(I) G(J; M) G(I+J; M) be . The first purpose of this paper is to prove the following theorem. |
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