---
date: 2022-05-08 18:51
modification date: Sunday 8th May 2022 18:51:11
title: "fracture theorem"
aliases: [fracture theorem]
created: 2023-04-03T11:46
updated: 2023-04-03T11:47
---

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

---

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

---

# fracture theorem

![](2023-04-03-7.png)

For $\ZZ$:

![](attachments/Pasted%20image%2020220508185512.png)

\begin{tikzcd}
	\ZZ && {\QQ^\disc} \\
	&&& \in\Top\CRing \\
	{\AA_\ZZ = \displaystyle\prod_{p\in \abs{\spec \ZZ}}\ZZpadic} && {\AA_\QQ \cong \AA_\ZZ\tensor_\ZZ \QQ}
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-3, to=1-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXFpaIl0sWzIsMCwiXFxRUV5cXGRpc2MiXSxbMCwyLCJcXEFBX1xcWlogPSBcXHByb2Rfe3BcXGluIFxcYWJze1xcc3BlYyBcXFpafX1cXFpacGFkaWMiXSxbMiwyLCJcXEFBX1xcUVEgXFxjb25nIFxcQUFfXFxaWlxcdGVuc29yX1xcWlogXFxRUSJdLFszLDEsIlxcaW5cXFRvcFxcQ1JpbmciXSxbMCwxXSxbMiwzXSxbMSwzXSxbMCwyXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbMywwLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)


For an arbitrary abelian group:

\begin{tikzcd}
	G && {\AA_{\QQ}(G) \da G\tensor_\ZZ \QQ} \\
	\\
	{\AA_{\ZZ}(G) \da \displaystyle\prod_{p\in \abs{\spec \ZZ}} G_p} && {\AA_{\ZZ}(G) \tensor_\ZZ \QQ}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHIl0sWzAsMiwiXFxBQV97XFxaWn0oRykgXFxkYSBcXGRpc3BsYXlzdHlsZVxccHJvZF97cFxcaW4gXFxhYnN7XFxzcGVjIFxcWlp9fSBHX3AiXSxbMiwwLCJcXEFBX3tcXFFRfShHKSBcXGRhIEdcXHRlbnNvcl9cXFpaIFxcUVEiXSxbMiwyLCJcXEFBX3tcXFpafShHKSBcXHRlbnNvcl9cXFpaIFxcUVEiXSxbMCwyXSxbMCwxXSxbMSwzXSxbMiwzXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)


# In spaces

![](attachments/Pasted%20image%2020220508193239.png)

# In spectra

![](attachments/Pasted%20image%2020220508190333.png)