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Cotangent spaces and separating re-embeddings
Journal of Algebra and Its Applications ( IF 0.5 ) Pub Date : 2021-06-22 , DOI: 10.1142/s0219498822501882
Martin Kreuzer 1 , Le Ngoc Long 2 , Lorenzo Robbiano 3
Affiliation  

Given an affine algebra R=P/I, where P=K[x1,,xn] is a polynomial ring over a field K and I is an ideal in P, we study re-embeddings of the affine scheme Spec(R), i.e. presentations RP/I such that P is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials fi in the ideal I which are coherently separating in the sense that they are of the form fi=zigi with an indeterminate zi which divides neither a term in the support of gi nor in the support of fj for ji. 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.



中文翻译:

余切空间和分离重新嵌入

给定一个仿射代数R=/, 在哪里=ķ[X1,,Xn]是域上的多项式环ķ是一个理想的 ,我们研究仿射方案的重新嵌入规格(R),即演示文稿R'/'这样'是较少不定项的多项式环。为了找到这样的重新嵌入,我们使用多项式F一世在理想中它们是连贯地分开的,因为它们是形式F一世=z一世-G一世带有不确定的z一世它既不划分一个术语来支持G一世也不支持Fj为了j一世. 这样的多项式集合的可能数量由 Gröbner fan 控制. 余切空间的维数 R在一个ķ-线性最大理想是嵌入维数的下界,如果我们找到对应于这个界的相干分离多项式,我们知道我们已经确定了嵌入维数R并找到了一个最佳的重新嵌入。

更新日期:2021-06-22
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