## 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 $R: 𝒪X→𝒪Y such that if f1,f2 ∈𝒪X then R(f1f2 ) =R(f1 R(f2).$ A derivation of ${𝒪}_{X}$ is an $𝔽$-linear map $∂: 𝒪X→𝒪X such that if f1,f2 ∈𝒪X then ∂(f1f2 ) = f1 ∂(f2) + ∂(f1) f2.$

Let $X$ be a space and let $𝒪X= {functions f:X→𝔽}$ 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 $if f1,f2 ∈𝒪X then x(f1f2 ) =x(f1) x(f2).$ Hence, $X=Hom𝔽-alg (𝒪X,𝔽) .$ 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 $η:𝒪X →𝔽 such that η(f1 f2) = f1(x) η(f2) + η(f1) f2(x),$ for ${f}_{1},{f}_{2}\in {𝒪}_{X}$. The tangent bundle to $X$ is $T(X) = Hom𝔽-alg( 𝒪X, 𝔽[t]/ ⟨t2⟩) with T(X) t= ↓ t=0 X$ If $\gamma \in T\left(X\right)$ and $\xi :{𝒪}_{X}\to 𝔽$ and $\eta :{𝒪}_{X}\to 𝔽$ are such that $γ=ξ+tη then ξ(f1 f2) = ξ(f1) ξ(f2) and η(f1 f2) = ξ(f1) η(f2) + η(f1) ξ(f2)$ so that, by identifying $\xi$ with a point $x\in X$, $η(f1 f2) = f1(x) η(f2) + η(f1) f2(x) ,for f1,f2 ∈𝒪X,$ 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 $∂:𝒪X →𝒪X satisfies ∂(f1f2 ) = f1 ∂(f2) + ∂(f1) f2,$ for ${f}_{1},{f}_{2}\in {𝒪}_{X}$, and ${derivations of 𝒪X} = {vector fields on X} = {sections of T(X)} .$

If $\phi :X\to Y$ is a morphism then $dφ: Tx(X) ⟶ Tφ(x) (Y) η ⟼ η∘φ* and d(φ∘ψ) = dφ∘dψ$ is a generalization of the chain rule from calculus. (Proof: Let $\eta \in {T}_{x}\left(X\right)$ and $f\in {𝒪}_{Z}$. Since $d(ψ∘φ) (η)(f) = ( η(ψ∘φ) *f) = η(fψφ)$ and $(dψ∘dφ) η(f) = dψ(η φ*)(f) = (ηφ* ψ*)(f) = η(φ*( ψ*f)) = η(fψφ).$ it follows that $d\left(\phi \circ \psi \right)=d\phi \circ d\psi$. Since $dφ(η) (f1f2) = ηφ* (f1f2) = η( φ*(f1) φ*(f2) ) = (φ*f1) (x) ηφ* (f2) +ηφ* (f1) (φ*f2) (x) = f1(φ(x)) dη(f2+ dη(f1) f2(φ(x) )$ 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 $0⟶𝒪X ⟶d ΩX1 ⟶d ΩX2 ⟶d ⋯ ,$ where the $p$-forms on $X$ are the elements of $ΩXp = Λp( ΩX1) , ΩX1 =I/I2, where I= ker( 𝒪X⊗ 𝒪X ⟶ 𝒪X f1⊗f2 ⟼ f1f2 )$ and $d$ is the unique antiderivationWHAT DOES THIS WORD MEAN? of degree 1 extending $d: 𝒪X ⟶ ΩX1 f ⟼ f⊗1- 1⊗f$ Then ${𝒪}_{X}$ acts on ${\Omega }_{X}^{1}$ by $f( ∑gi⊗ fi ) = ∑fgi⊗ hi = ∑gi⊗fhi mod I2 ,$ for $f\in {𝒪}_{X}$ and $\sum {g}_{i}\otimes {h}_{i}\in I$. As ${𝒪}_{X}$-modules $Hom𝒪X( ΩX1, 𝒪X) ⟶∼ Der(𝒪X) d ⟼ ωd$ Note that, if $\omega \in {\mathrm{Hom}}_{{𝒪}_{X}}\left({\Omega }_{X}^{1},{𝒪}_{X}\right)$ then ${𝒪}_{X}\stackrel{d}{\to }{\Omega }_{X}^{1}\stackrel{\omega }{\to }{𝒪}_{X}$ and $f1⋅ (ωd) (f2) + (ωd) (f1) ⋅f2 = f1⋅ ω(f2⊗1 -1⊗f2) + ω(f1⊗1 -1⊗f1) f2 = ω(f1f2 ⊗1-f1 ⊗f2) + ω(f1⊗ f2-1⊗ f1f2) = ω(f1f2 ⊗1-1⊗f1f2) = (ωd)(f1 f2).$ If ${\Omega }_{X}^{1}$ is a reflexiveWHAT DOES THIS WORD MEAN? ${𝒪}_{X}$-module then $ΩX1 = Hom𝒪X( Der(𝒪X), 𝒪X) .$

Example. Let ${𝒪}_{X}=𝔽\left[{x}_{1},\dots ,{x}_{n}\right]$ so that $X={𝔽}^{n}$. If $v\in {𝔽}^{n}$ then the homomorphism $𝒪X ⟶ 𝔽[t] ⟨t2⟩ spacerspacerspa f ⟼ f(x+tv) = f(x)+ t∂v |x$ defines the derivative ${\partial }_{v}$ in the direction $v$, $∂v= ∑i=1n vi ∂xi ∂xi , if v=(v1, …,vn) .$ Then

• $\left\{\text{sections of}\phantom{\rule{0.3em}{0ex}}T\left(X\right)\right\}$ has ${𝒪}_{X}$-basis $\left\{\frac{\partial \phantom{{x}_{i}}}{\partial {x}_{1}},\dots ,\frac{\partial \phantom{{x}_{i}}}{\partial {x}_{n}}\right\}$, and
• $\left\{\text{sections of}\phantom{\rule{0.3em}{0ex}}{T}^{*}\left(X\right)\right\}$ has ${𝒪}_{X}$-basis $\left\{d{x}_{1},\dots ,d{x}_{n}\right\}$ (HOW DO WE IDENTIFY THESE AS ELEMENTS OF $I/{I}^{2}$?)
and ${\Omega }_{X}^{p}=\left\{d{x}_{{i}_{1}}\wedge \cdots \wedge d{x}_{{i}_{p}}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}1\le {i}_{1}<\cdots <{i}_{p}\le n\right\}$ with $df= ∂v= ∑i=1n ∂f ∂xi dxi and d(f( dxi1 ∧⋯∧ dxip )) = df∧ dxi1 ∧⋯∧ dxip ,$ for $f\in 𝔽\left[{x}_{1},\dots ,{x}_{n}\right]$.

## 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 ${T}_{a}\left(f\right):\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)$ is injective.

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

$\text{if}\phantom{\rule{1em}{0ex}}a\in X\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{T}_{a}f:\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)\phantom{\rule{1em}{0ex}}\text{is injective.}$

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

${T}_{a}f:\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)\phantom{\rule{1em}{0ex}}\text{is an isomorphism.}$

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

$\text{if}\phantom{\rule{1em}{0ex}}a\in X\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{T}_{a}f:\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)\phantom{\rule{1em}{0ex}}\text{is an isomorphism.}$

An submersion at a is a morphism $X\stackrel{f}{⟶}Y$ of varieties such that ${T}_{a}\left(f\right):\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)$ is surjective.

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

$\text{if}\phantom{\rule{1em}{0ex}}a\in X\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{T}_{a}f:\phantom{\rule{0.2em}{0ex}}{T}_{a}\left(X\right)⟶{T}_{f\left(a\right)}\left(Y\right)\phantom{\rule{1em}{0ex}}\text{is surjective.}$

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

$\begin{array}{cccccc}f:& X& \stackrel{j}{⟶}& X×Y& \stackrel{p{r}_{2}}{⟶}& Y\\ & x& ⟼& \left(x,f\left(x\right)\right)& & \\ & & & \left(x,y\right)& ⟼& y\end{array}$

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, ??????