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created: 2022-04-05T23:42
updated: 2023-12-19T15:28
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date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: local complete intersection
aliases: [local complete intersection, lci, complete intersections]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #AG
- Refs:
	- #todo/add-references
- Links:
	- [Lefschetz hyperplane theorem](Lefschetz%20hyperplane%20theorem)

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# local complete intersection

For $X$ a [finite type](Unsorted/finite%20type.md)  $k\dash$algebra, $X$ is a **local complete intersection** if there is a covering by distinguished opens $D(g_i)$ such that the local rings $X\localize{g_i}$ admit presentations of the form $X\localize{g_i} = \kxn/\gens{f_1, \cdots, f_d}$ with $\codim(X) = d$.

For a Noetherian local ring $A$, its completion $A\complete{\mfm_A}$ is the quotient of a [regular local ring](Unsorted/regular%20ring.md) by a an ideal generated by a [regular sequence](regular%20sequence). Equivalently, if $A$ is a complete Noetherian local ring, there is a resolution $M\to R\to A$ where $R$ is a regular local ring and $M$ is generated by a regular sequence.

For an algebraic variety $V \subseteq \PP^n$, $I(V)$ is generated by exactly $d\da \codim V$ elements so that $V$ is the intersection of exactly $D$ hypersurfaces.

Idea: they can be defined using the minimal number of relations.

![](attachments/Pasted%20image%2020220417014544.png)
# derived version

![](2023-12-19-1.png)
See [regular sequence](regular%20sequence.md).

Characterization in terms of the [cotangent complex](cotangent%20complex.md):
![](2023-12-19-2.png)