---
date: 2022-01-24 12:08
modification date: Monday 24th January 2022 15:43:55
title: adic completion
aliases: ["adic completion", "completion", "complete", "complete ring", "algebraic completion", "formal neighborhoods are completions"]

---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# adic completion

The ur-example: [p-adic integers](Unsorted/p-adic.md).

![](attachments/Pasted%20image%2020220508184210.png)

Idea: for schemes, completion at a point is a formal neighborhood.

# Completion wrt a filtration
For $(M, \Fil^*)$ a (descending) filetered module, one forms the completion as
$$
M = \Fil^0 M \supseteq \Fil^1 M \supseteq \cdots \implies \hat{M} \da M\complete{\Fil^*} \da \inverselim_{n}\, {M \over \Fil^n M}
$$
If $\Fil^*$ is a terminating filtration, the result is a topological module.
# adic completion
![](attachments/Pasted%20image%2020220124120838.png)

So
$$
R\complete{I} = \inverselim_n R/I^n
$$

Usually involves localizing first:
![](attachments/Pasted%20image%2020220124225812.png)
![](attachments/Pasted%20image%2020220407235134.png)

# Results

- [Hensel's Lemma](Unsorted/Hensel's%20Lemma.md) applies to complete rings.
- Coincides with [Cauchy completion](Unsorted/Cauchy%20completion.md) when $R$ has a metric induced by a [nonarchimedean absolute value](Unsorted/absolute%20value.md).


# Examples

- $\ZZ\complete{\gens{p}} = \ZZ\pcomplete$ is the [p-adic integers](Unsorted/p-adic.md).
- $k[x_1,\cdots, x_n]\complete{\gens{x_1,\cdots, x_n}} = k\fps{x_1, \cdots, x_n}$ are multivariate formal power series.
# Exercises

- Show that the completion of a [Noetherian](Unsorted/Noetherian.md) ring $R$ is flat as an $R\dash$module.
- Show that completion can be computed by [extension of scalars](Unsorted/base%20change.md) as
$$
M \complete{I} \cong M \tensor_R R\complete{I}
$$
- Use this to show $(R/J)\complete{I} \cong R\complete{I}/J\complete{I}$.
- ![](attachments/Pasted%20image%2020220407235059.png)