论文标题
cotangent空间和分开重新安装
Cotangent spaces and separating re-embeddings
论文作者
论文摘要
给定仿射代数$ r = p/i $，其中$ p = k [x_1，\ dots，x_n] $是$ k $的多项式戒指，$ k $，$ i $是$ p $的理想之处，我们研究了affine offine offine offine offine offine offine offine offine offine offine $'不确定的响起。 To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i= z_i - g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$.此类多项式的可能数量显示为$ i $的Gröbner粉丝。 $ r $在$ k $ - 线性最大理想下的cotangent空间的尺寸是嵌入尺寸的下限，如果我们发现与此相对应的多项式分离的多项式分开，我们知道我们已经确定$ r $的嵌入尺寸并找到了一个最佳重新安装。
Given an affine algebra $R=P/I$, where $P=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme ${\rm Spec}(R)$, i.e., presentations $R \cong P'/I'$ such that $P'$ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i= z_i - g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of $I$. The dimension of the cotangent space of $R$ at a $K$-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of $R$ and found an optimal re-embedding.
