---
date: 2022-04-05 23:42
modification date: Thursday 7th April 2022 22:40:37
title: "Zariski tangent space"
aliases: [tangent space]
---

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

---
- Tags: 
	- #AG/schemes 
- Refs:
	- #todo/add-references
- Links:
	- [singular](Unsorted/smooth%20points.md)

---

# Zariski tangent space


See [dual numbers](dual%20numbers), [closed point](Unsorted/special%20fiber.md)

![](attachments/Pasted%20image%2020220407224036.png)
![](attachments/Pasted%20image%2020220407224315.png)
![](attachments/Pasted%20image%2020220407224326.png)

![](attachments/Pasted%20image%2020220415131553.png)

Equivalent definitions:

- For manifolds:
	- $\T_p M = \ts{ \gamma\in C^\infty(I, M) \st \gamma(0) = p}/\sim$ where $\gamma \sim \eta \iff \gamma'(0) = \eta'(0)$.
	- $\T_p M = (\mfm_p/\mfm_p^2)\dual \da \kmod(\mfm_p/\mfm_p^2, k)$ where $\mfm_p = \ts{f\in C^\infty(M; k) \st f(p) = 0} \in \mspec C^\infty(M; k)$
	- $\T_p M = \Der_{\kmod}(C^\infty(M; k); k)$
- For affine schemes $M = \spec R$ and $\mfm \in \mspec R$ a closed point:
	- $\T_\mfm M = \kalg(R, k[\eps])$
	- $\T_\mfm M = \Der_{\kmod}(R; k)$ -- letting $\proj_\mfm: R\to R/\mfm$, these must satisfy $D(ab) = \proj_\mfm(a) D(b) + \proj_\mfm(b) D(a)$.
- For general schemes: todo. Use the stalk $R_\mfm$ and embed $\mfm \injects R_\mfm$.

- Can be defined by taking defining equations $\ts{f_i}_{i=1}^r \subseteq \kx{n}$, taking the matrix of partials $\tv{\dd{f_i}{x_j}}$, which evaluating at a point $p$ yields a map $Df: \FF^n \to \FF^r$ where $\FF$ is the [residue field](residue%20field) at $p$.
	- Then define the **Zariski tangent space** as $\ker D_f$.