Cotangent spaces and separating re-embeddings

TL;DR

This paper investigates re-embeddings of affine schemes using coherently separating polynomials, linking their existence to the Gröbner fan, and provides methods to determine the embedding dimension via cotangent spaces.

Contribution

It introduces a new approach to find optimal re-embeddings of affine schemes using coherently separating polynomials and relates this to the Gröbner fan and cotangent space dimensions.

Findings

Re-embeddings are governed by the Gröbner fan of the ideal.

Cotangent space dimension provides a lower bound for embedding dimension.

Identifies conditions for optimal re-embeddings using separating polynomials.

Abstract

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 Gr\"obner 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.