Regular embedding

In algebraic geometry, a closed immersion $i: X \hookrightarrow Y$ of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of $X \cap U$ is generated by a regular sequence of length r.

For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If $\operatorname{Spec}B$ is regularly embedded into a regular scheme, then B is a complete intersection ring.

单词	Regular embedding
释义	Regular embedding 英语百科 Regular embedding In algebraic geometry, a closed immersion $i: X \hookrightarrow Y$ of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of $X \cap U$ is generated by a regular sequence of length r. For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If $\operatorname{Spec}B$ is regularly embedded into a regular scheme, then B is a complete intersection ring.
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