Tangent spaces and Differential forms

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 7 December 2011

Spaces

An $𝔽$-algebra is a ring ${𝒪}_X$ that is also a vector space over $𝔽$. A homomorphism of algebras is an $𝔽$-linear map $$R:{𝒪}_X\to {𝒪}_Y\text{such that}\text{if}{f}_1,f_2\in {𝒪}_X\text{then}R\left(f_1{f}_2\right)=R\left(f_{1}R\right(f_2).$$ A derivation of ${𝒪}_X$ is an $𝔽$-linear map $$\partial :{𝒪}_X\to {𝒪}_X\text{such that}\text{if}{f}_1,f_2\in {𝒪}_X\text{then}\partial \left(f_1{f}_2\right)=f_1\partial \left(f_2\right)+\partial \left(f_1\right)f_2.$$

Let $X$ be a space and let $${𝒪}_X=\{\text{functions}f:X\to 𝔽\}$$ be the algebra of functions on $X$. If $x\in X$ and $f\in {𝒪}_X$ let $x\left(f\right)=f\left(x\right)$ so that $$\text{if}{f}_1,f_2\in {𝒪}_X\text{then}x\left(f_1{f}_2\right)=x\left(f_1\right)x\left(f_2\right).$$ Hence, $$X={\mathrm{Hom}}_{𝔽\text{-alg}}({𝒪}_X,𝔽).$$ A morphism $\phi :X\to Y$ corresponds to the morphism ${\phi }^{*}:{𝒪}_Y\to {𝒪}_X$ given by ${\phi }^{*}\left(f\right)=f\circ \phi$, for $f\in {𝒪}_Y$.

The tangent bundle

Let $X$ be a space with ${𝒪}_X$ the ring of functions on $X$. Let $x\in X$. A tangent vector to $X$ at $x$ is a linear map $$\eta :{𝒪}_X\to 𝔽\text{such that}\eta \left(f_1{f}_2\right)=f_1\left(x\right)\eta \left(f_2\right)+\eta \left(f_1\right)f_2\left(x\right),$$ for $f_1,f_2\in {𝒪}_X$. The tangent bundle to $X$ is $$T\left(X\right)={\mathrm{Hom}}_{𝔽\text{-alg}}({𝒪}_X,𝔽[t]/⟨t^2⟩)\text{with}\begin{array}{c}T\left(X\right)\\ \phantom{t=}\downarrow t=0\\ X\end{array}$$ If $\gamma \in T\left(X\right)$ and $\xi :{𝒪}_X\to 𝔽$ and $\eta :{𝒪}_X\to 𝔽$ are such that $$\gamma =\xi +t\eta \text{then}\xi \left(f_1{f}_2\right)=\xi \left(f_1\right)\xi \left(f_2\right)\text{and}\eta \left(f_1{f}_2\right)=\xi \left(f_1\right)\eta \left(f_2\right)+\eta \left(f_1\right)\xi \left(f_2\right)$$ so that, by identifying $\xi$ with a point $x\in X$, $$\eta \left(f_1{f}_2\right)=f_1\left(x\right)\eta \left(f_2\right)+\eta \left(f_1\right)f_2\left(x\right),\text{for}{f}_1,f_2\in {𝒪}_X,$$ and $\eta$ is a tangent vector to $X$ at $x$. A vector field is a section $\partial$ of $T\left(X\right)$, i.e. a choice of a tangent vector at each point $x\in X$. Hence $$\partial :{𝒪}_X\to {𝒪}_X\text{satisfies}\partial \left(f_1{f}_2\right)=f_1\partial \left(f_2\right)+\partial \left(f_1\right)f_2,$$ for $f_1,f_2\in {𝒪}_X$, and $$\left\{\text{derivations of}{𝒪}_X\right\}=\left\{\text{vector fields on}X\right\}=\left\{\text{sections of}T\right(X\left)\right\}.$$

If $\phi :X\to Y$ is a morphism then $$\begin{array}{cccc}d\phi :& T_x\left(X\right)& ⟶& T_{\phi \left(x\right)}\left(Y\right)\\ & \eta & ⟼& \eta \circ {\phi }^{*}\end{array}\text{and}d(\phi \circ \psi )=d\phi \circ d\psi$$ is a generalization of the chain rule from calculus. (Proof: Let $\eta \in T_x\left(X\right)$ and $f\in {𝒪}_Z$. Since $$d(\psi \circ \phi )\left(\eta \right)\left(f\right)=\left({\eta (\psi \circ \phi )}^{*}f\right)=\eta \left(f\psi \phi \right)$$ and $$(d\psi \circ d\phi )\eta \left(f\right)=d\psi \left(\eta {\phi }^{*}\right)(f)=(\eta {\phi }^{*}{\psi }^{*}\left)\right(f)=\eta ({\phi }^{*}\left({\psi }^{*}f\right))=\eta (f\psi \phi ).$$ it follows that $d(\phi \circ \psi )=d\phi \circ d\psi$. Since $$\begin{array}{rl}d\phi \left(\eta \right)\left(f_1{f}_2\right)& =\eta {\phi }^{*}\left(f_1{f}_2\right)=\eta \left({\phi }^{*}\right(f_1\left){\phi }^{*}\right(f_2\left)\right)\\ & =\left({\phi }^{*}{f}_1\right)\left(x\right)\eta {\phi }^{*}\left(f_2\right)+\eta {\phi }^{*}\left(f_1\right)\left({\phi }^{*}{f}_2\right)\left(x\right)& & =f_1\left(\phi \right(x\left)\right)d\eta (f_2+d\eta (f_1\left)f_2\right(\phi \left(x\right))\end{array}$$ it follows $d\eta$ is a tangent vector to $Y$ at $\phi \left(x\right)$. Thus the map $d\phi$ is well defined.

Differential forms

Let $X$ be a space with ring of functions ${𝒪}_X$. The de Rham cohomology of $X$ is the cohomology of the complex $$0⟶{𝒪}_X\stackrel{d}{⟶}{\Omega }_X^1\stackrel{d}{⟶}{\Omega }_X^2\stackrel{d}{⟶}\cdots ,$$ where the $p$-forms on $X$ are the elements of $${\Omega }_X^p={\Lambda }^p\left({\Omega }_X^1\right),{\Omega }_X^1=I/I^2,\text{where}I=\ker \left(\begin{array}{ccc}{𝒪}_X\otimes {𝒪}_X& ⟶& {𝒪}_X\\ f_1\otimes f_2& ⟼& f_1{f}_2\end{array}\right)$$ and $d$ is the unique antiderivationWHAT DOES THIS WORD MEAN? of degree 1 extending $$\begin{array}{cccc}d:& {𝒪}_X& ⟶& {\Omega }_X^1\\ & f& ⟼& f\otimes 1-1\otimes f\end{array}$$ Then ${𝒪}_X$ acts on ${\Omega }_X^1$ by $$f(\sum g_i\otimes f_i)=\sum f{g}_i\otimes h_i=\sum g_i\otimes f{h}_i\mathrm{mod}{I}^2,$$ for $f\in {𝒪}_X$ and $\sum g_i\otimes h_i\in I$. As ${𝒪}_X$-modules $$\begin{array}{ccc}{\mathrm{Hom}}_{{𝒪}_X}({\Omega }_X^1,{𝒪}_X)& \stackrel{\sim }{⟶}& \mathrm{Der}\left({𝒪}_X\right)\\ d& ⟼& \omega d\end{array}$$ Note that, if $\omega \in {\mathrm{Hom}}_{{𝒪}_X}({\Omega }_X^1,{𝒪}_X)$ then ${𝒪}_X\stackrel{d}{\to }{\Omega }_X^1\stackrel{\omega }{\to }{𝒪}_X$ and $$\begin{array}{rl}{f}_1\cdot \left(\omega d\right)\left(f_2\right)+\left(\omega d\right)\left(f_1\right)\cdot f_2& =f_1\cdot \omega (f_2\otimes 1-1\otimes f_2)+\omega (f_1\otimes 1-1\otimes f_1)f_2\\ & =\omega (f_1{f}_2\otimes 1-f_1\otimes f_2)+\omega (f_1\otimes f_2-1\otimes f_1{f}_2)\\ & =\omega (f_1{f}_2\otimes 1-1\otimes f_1{f}_2)=\left(\omega d\right)\left(f_1{f}_2\right).\end{array}$$ If ${\Omega }_X^1$ is a reflexiveWHAT DOES THIS WORD MEAN? ${𝒪}_X$-module then $${\Omega }_X^1={\mathrm{Hom}}_{{𝒪}_X}\left(\mathrm{Der}\right({𝒪}_X),{𝒪}_X).$$

Example. Let ${𝒪}_X=𝔽[x_1,\dots ,x_n]$ so that $X={𝔽}^n$. If $v\in {𝔽}^n$ then the homomorphism $$\begin{array}{ccc}{𝒪}_X& ⟶& \frac{𝔽\left[t\right]}{⟨t^2⟩}\phantom{\mathrm{spacerspacerspa}}\\ f& ⟼& f(x+tv)=f\left(x\right)+t{\partial }_v{|}_x\end{array}$$ defines the derivative ${\partial }_v$ in the direction $v$, $${\partial }_v=\sum _{i=1}^n{v}_i\frac{\partial \phantom{x_i}}{\partial x_i},\text{if}v=(v_1,\dots ,v_n).$$ Then

• $\left\{\text{sections of}T\right(X\left)\right\}$ has ${𝒪}_X$-basis $\{\frac{\partial \phantom{x_i}}{\partial x_1},\dots ,\frac{\partial \phantom{x_i}}{\partial x_n}\}$, and
• $\left\{\text{sections of}{T}^{*}\right(X\left)\right\}$ has ${𝒪}_X$-basis $\{d{x}_1,\dots ,d{x}_{\mathrm{n}}\}$ (HOW DO WE IDENTIFY THESE AS ELEMENTS OF $I/I^2$?)

and ${\Omega }_X^p=\{d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p}|1\le i_1<\cdots <i_p\le n\}$ with $$df={\partial }_v=\sum _{i=1}^n\frac{\partial f}{\partial x_i}d{x}_i\text{and}d\left(f\right(d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p}\left)\right)=df\wedge d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p},$$ for $f\in 𝔽[x_1,\dots ,x_n]$.

Bourbaki, Varietes Differentielles et Analytiques §5.7-5.10.

Tangent spaces, immersions, submersions and étale morphsisms.

Let $X$ be a variety and $a\in X$.

An immersion at a is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that $T_a\left(f\right):T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)$ is injective.

An immersion is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that

$\text{if}a\in X\text{then}{T}_{a}f:T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)\text{is injective.}$

A local isomorphism at a, or étale morphism at a, is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that

$T_{a}f:T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)\text{is an isomorphism.}$

An étale morphism is a morphism $X\stackrel{f}{⟶}Y$ such that

$\text{if}a\in X\text{then}{T}_{a}f:T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)\text{is an isomorphism.}$

An submersion at a is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that $T_a\left(f\right):T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)$ is surjective.

An submersion is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that

$\text{if}a\in X\text{then}{T}_{a}f:T_a\left(X\right)⟶T_{f\left(a\right)}\left(Y\right)\text{is surjective.}$

HW: If $X\stackrel{f}{⟶}Y$ is a morphism then

$\begin{array}{cccccc}f:& X& \stackrel{j}{⟶}& X\times Y& \stackrel{p{r}_2}{⟶}& Y\\ & x& ⟼& (x,f\left(x\right))& & \\ & & & (x,y)& ⟼& y\end{array}$

with $j$ an immersion and $p{r}_2$ a submersion.

Notes and References

This page is influenced by Macdonald [CSM, ???] and [KL, ???]. The fundamental idea that a space (collection of points in space) is characterized by its ring of functions revolutionized mathematical thought in the 20th century. Significant exploration of this idea occurred in the development of functional analysis (Gelfand school) and algebraic geometry (Grothendieck school).

References

[CSM] R. Carter, G. Segal and I.G. Macdonald, Lectures on Lie groups and Lie algebras, London Mathematical Society Lecture Notes, ??????

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