论文标题
相对自由的仿射代数在Noetherian环上的代表性
Representability of relatively free affine algebras over a Noetherian ring
论文作者
论文摘要
多年来，关于（非共同）多项式的T理想的问题已经出现。但是，在评估矩阵亚综合总代综合物中的非中性多项式时，人们通常在确定多项式的特定评估方面几乎没有控制。克服特征上难度0的一种方法是减少多线性多项式并利用对称群的表示理论。但是该技术在特征$ p> 0 $中不可用。 成功的一种替代方法是``远足''一种多项式的过程，其中人们在几个阶段中专门将其不确定的人进行了专业，以获得包含Capelli多项式的多项式，以控制其评估。该方法用于在Specht对阳性特征领域的仿射代数的猜想证明中使用。在本文中，我们将进一步远足到非均匀的多项式，以适用于代表性问题。凯默（Kemer）在1988年证明，每个相对自由的pi代数在无限领域都可以代表。在2010年，本文的第一作者更普遍地证明，每个相对自由的PI代数都超过了任何可交换的Noetherian Unitean Ring，这是可以代表的。我们基于有限领域的非均匀多项式远足，基于远足的非均匀多项式提供了不同的，完整的证明。然后，我们使用Noetherian在T-Ideals上诱导Noetherian的诱导，从而在Noetherian的交换环上获得完整的结果。 大部分证明是用于积极特征的基础场的情况。在这里，远足的用法比证明Specht的猜想更为直接，但当基本环是有限的时，必须考虑非均匀的多项式，这需要克服某些困难。 在附录中，我们展示了如何适应远足以证明具有参与性的版本以及各种分级和非社交定理。
Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and utilizing the representation theory of the symmetric group. But this technique is unavailable in characteristic $p>0$. An alternative method, which succeeds, is the process of ``hiking'' a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial that contains Capelli polynomials, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper we develop hiking further to nonhomogeneous polynomials, to apply to the representability question. Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In the appendix we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems.
