论文标题
多元塞雷猜想环
The multivariate Serre conjecture ring
论文作者
论文摘要
众所周知，对于任何可交换的统一戒指$ \ mathbf {r} $，Serre猜测环$ \ Mathbf {r} \ langle x \ rangle $，即，Univariate polynomial ring $ \ nathbf {x] $ nirivariate polynomial $ nimic prolyq proll le srain ins bull leull nimials bull leull nivariate polynomial ring ull ull ull nivariate polly nimial is bull leull 1 $如果是$ \ Mathbf {r} $。因此，通过归纳定义$ \ mathbf {r} \ langle x_1，\ ldots，x_n \ rangle：=（\ Mathbf {r} \ langle x_1，\ ldots，x_ {n-1} x_1，\ ldots，x_n \ rangle $是krull dimension $ \ leq 1 $的bézout域，如果是$ \ mathbf {r} $。 $ \ langle x_1，\ ldots，x_n \ rangle $是一个bézout域，当$ \ m \ m \ mthbf {r} $是krull dimension $ \ leq 1 $是Brewer和Costa的基础，这一事实是一个bézout域名然后，算术环有限地生成的投影$ \ mathbf {r} [x_1，\ dots，x_n] $ - 模块被扩展。它也是在词典订单案例中Gröbner环的猜想证明的关键，即，对于任何估值域而言，Krull Dimension $ \ leq 1 $的任何估值域$ \ Mathbf {r} $ \ dots，x_n] $，理想的$ \ operatotorname {lt}（i）$由$ i $的元素相对于词典单一订单生成的元素生成。由于环$ \ Mathbf {r} \ langle x_1，\ ldots，x_n \ rangle $也可以直接定义为多变量多项式环$ \ mathbf {r} [x_1，\ dots，\ dots，x_n] $在polynomials上，该级别的coleff prodial proceffercipers with with with with prodial procecroprice with prodial with with with prodialsials的定位， $ x_1 <x_2 <\ cdots <x_n $是$ 1 $，我们建议概括这样一个事实：$ \ mathbf {r} \ langle x_1，\ ldots，x_n \ rangle $是krull dimension $ \ leq 1 $ if $ \ m ranical bor belication $ bere be y mathbf} $ bere buull dimension $ be}在理性案例中，格布纳环的猜想。
It is well-known that for any commutative unitary ring $\mathbf{R}$, the Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a Bézout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. Consequently, defining by induction $\mathbf{R}\langle X_1,\ldots,X_n \rangle:=(\mathbf{R}\langle X_1,\ldots,X_{n-1}\rangle)\langle X_n\rangle$, the ring $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a Bézout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. The fact that $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a Bézout domain when $\mathbf{R}$ is a valuation domain of Krull dimension $\leq 1$ was the cornerstone of Brewer and Costa's theorem stating that if $\mathbf{R}$ is a one-dimensional arithmetical ring then finitely generated projective $\mathbf{R}[X_1,\dots,X_n]$-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain $\mathbf{R}$ of Krull dimension $\leq 1$, any $n \in \mathbb{N}_{>0}$, and any finitely generated ideal $I$ of $\mathbf{R}[X_1, \dots, X_n]$, the ideal $\operatorname{LT}(I)$ generated by the leading terms of the elements of $I$ with respect to the lexicographic monomial order is finitely generated. Since the ring $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ can also be defined directly as the localization of the multivariate polynomial ring $\mathbf{R}[X_1,\dots,X_n]$ at polynomials whose leading coefficients according to the lexicographic monomial order with $X_1<X_2<\cdots<X_n$ is $1$, we propose to generalize the fact that $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ is a Bézout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$ to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case.
