Normal transversality and uniform bounds
نویسنده
چکیده
Let A be a commutative ring. A graded A-algebra U = ⊕n≥0Un 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 = ⊕n≥0Fn is a standard U -module if F is generated as an U -module by the elements of F0, that is, Fn = UnF0 for all n ≥ 0. In particular, Fn = U1Fn−1 for all n ≥ 1. Given I, J , two ideals of A, we consider the following standard algebras: the Rees algebra of I, R(I) = ⊕n≥0I t = A[It] ⊂ A[t], and the multi-Rees algebra of I and J , R(I, J) = ⊕n≥0(⊕p+q=nI Juv) = A[Iu, Jv] ⊂ A[u, v]. Consider the associated graded ring of I, G(I) = R(I) ⊗ A/I = ⊕n≥0I /I, and the multi-associated graded ring of I and J , G(I, J) = R(I, J) ⊗ A/(I + J) = ⊕n≥0(⊕p+q=nI J/(I + J)IJ). We can always consider the tensor product of two standard A-algebras U = ⊕p≥0Up and V = ⊕q≥0Vq as an standard A-algebra with the natural grading U ⊗ V = ⊕n≥0(⊕p+q=nUp ⊗ Vq). If M is an A-module, we have the standard modules: the Rees module of I with respect to M , R(I;M) = ⊕n≥0I Mt =M [It] ⊂M [t] (a standard R(I)-module), and the multi-Rees module of I and J with respect to M , R(I, J ;M) = ⊕n≥0(⊕p+q=nI JMuv) =M [Iu, Jv] ⊂M [u, v] (a standard R(I, J)module). Consider the associated graded module of I with respect toM , G(I;M) = R(I;M)⊗A/I = ⊕n≥0I M/IM (a standard G(I)-module), and the multi-associated graded module of I and J with respect to M , G(I, J ;M) = R(I, J ;M)⊗A/(I + J) = ⊕n≥0(⊕p+q=nI JM/(I + J)IJM) (a standard R(I, J)-module). If U , V are two standard A-algebras and F is a standard U -module and G is a standard V -module, then F ⊗G = ⊕n≥0(⊕p+q=nFp ⊗Gq) is a standard U ⊗ V -module. Denote by π : R(I)⊗R(J ;M) → R(I, J ;M) and σ : R(I, J ;M) → R(I + J ;M) the natural surjective graded morphisms of standardR(I)⊗R(J)-modules. Let φ : R(I)⊗R(J ;M) → R(I + J ;M) be σ ◦ π. Denote by π : G(I)⊗ G(J ;M) → G(I, J ;M) and σ : G(I, J ;M) → G(I + J ;M) the tensor product of π and σ by A/(I + J); these are two natural surjective graded morphisms of standard G(I) ⊗ G(J)-modules. Let φ : G(I) ⊗ G(J ;M) → G(I + J ;M) be σ ◦ π. The first purpose of this note is to prove the following theorem:
منابع مشابه
An infinite-horizon maximum principle with bounds on the adjoint variable
We provide necessary optimality conditions for a general class of discounted infinitehorizon dynamic optimization problems. As part of the resulting maximum principle we obtain explicit bounds on the adjoint variable, stronger than the transversality conditions in Arrow–Kurz form. r 2005 Elsevier B.V. All rights reserved. JEL classification: C60; C61
متن کاملA Skein Approach to Bennequin Type Inequalities
We give a simple unified proof for several disparate bounds on Thurston–Bennequin number for Legendrian knots and self-linking number for transverse knots in R, and provide a template for possible future bounds. As an application, we give sufficient conditions for some of these bounds to be sharp.
متن کاملUniform and Non-uniform Bounds in Normal Approximation for Nonlinear Statistics
Let T be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry-Esseen type bounds for T are obtained. The bounds are best possible for many known statistics. Applications to U-statistic, multi-sample U-statistic, L-statistic, random sums, and functions of non-linear statistics are discussed.
متن کاملTransversal Families of Hyperbolic Skew-products
We study families of hyperbolic skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact. We introduce and employ probability measures (constructed from equilibrium measures on the natural extension)...
متن کاملPiecewise Linear Transversality
We prove transversality theorems for piecewise linear manifolds, maps and polyhedra. Our main result is that given two closed manifolds contained in a third, then one can be ambient isotoped until it is transversal to the other. This result is then extended to maps and polyhedra. The transversality theory for smooth manifolds was initiated by Thorn in his classical paper [8], and has been exten...
متن کامل