Spaces

Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu

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 $R:{𝒪}_X\to {𝒪}_Y$ such that

$R\left(f_1{f}_2\right)=R\left(f_1\right)R\left(f_2\right)\text{,}$ for $f_1,f_2\in {𝒪}_X$.

A derivation of ${𝒪}_X$ is an $𝔽$-linear map $\partial :{𝒪}_X\to {𝒪}_X$ such that

$\partial \left(f_1{f}_2\right)=f_1\partial \left(f_2\right)+\partial \left(f_1\right)f_2\text{,}$ for $f_1,f_2\in {𝒪}_X$.

Let $X$ be a space,

${𝒪}_X=\{\text{functions}f:X\to 𝔽\}$

the $𝔽$-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

$x\left(f_1{f}_2\right)=x\left(f_1\right)x\left(f_2\right)\text{,}$ for $f_1,f_2\in {𝒪}_X$.

Hence,

$X={\mathrm{Hom}}_{𝔽\text{-alg}}({𝒪}_X,𝔽)$.

A morphism $\varphi :X\to Y$ corresponds to the morphism ${\varphi }^{*}:{𝒪}_Y\to {𝒪}_X$ given by ${\varphi }^{*}\left(f\right)=f\circ \varphi$, for $f\in {𝒪}_Y$.

The tangent bundle

$\eta :{𝒪}_X\to 𝔽\text{such that}\eta \left(f_1{f}_2\right)=f_1\left(x\right)f_2)+\eta (f_1\left)f_2\right(x)$,

for $f_1,f_2\in {𝒪}_X$. The tangent bundle to $X$ is

$T\left(X\right)={\mathrm{Hom}}_{𝔽\text{-alg}}({𝒪}_X,𝔽[t]/\mathrm{langle}{t}^2\mathrm{rangle})$ 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

$\gamma =\xi +t\eta$ then $\xi \left(f_1{f}_2\right)=\xi \left(f_1\right)\xi \left(f_2\right)$ 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, 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

$\partial :{𝒪}_X\to {𝒪}_X$ 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

$\left\{\mathrm{derivations\; of}{𝒪}_X\right\}=\left\{\text{vector fields on}X\right\}=\left\{\text{sections of}T\right(X\left)\right\}$.

If $\varphi :X\to Y$ is a morphism then

$\begin{array}{cccc}d\varphi :& T_x\left(X\right)& ⟶& T_{\varphi \left(x\right)}\left(Y\right)\\ & \eta & \mapsto & \eta \circ {\varphi }^{*}\end{array}\text{and}d\left(\varphi \circ \psi \right)=d\varphi \circ d\psi$

is a generalization of the chain rule from calculus. If $\eta \in T_x\left(X\right)$ and $f\in {𝒪}_Z$ then

$d(\psi \circ \varphi )\left(\eta \right)\left(f\right)=\left(\eta \right(\psi \circ \varphi {)}^{*}f)=\eta (f\psi \varphi ),$

and

$(d\psi \circ d\varphi )\eta \left(f\right)=d\psi \left(\eta {\varphi }^{*}\right)\left(f\right)=\left(\eta {\varphi }^{*}{\psi }^{*}\right)\left(f\right)=\eta \left({\varphi }^{*}\right({\psi }^{*}f\left)\right)=\eta \left(f\psi \varphi \right).$

Note that

$\begin{array}{c}d\varphi \left(\eta \right)\left(f_1{f}_2\right)=\eta {\varphi }^{*}\left(f_1{f}_2\right)=\eta \left({\varphi }^{*}\right(f_1\left){\varphi }^{*}\right(f_2\left)\right)\\ =\left({\varphi }^{*}{f}_1\right)\left(x\right)\eta {\varphi }^{*}\left(f_2\right)+\eta {\varphi }^{*}\left(f_1\right)\left({\varphi }^{*}{f}_2\right)\left(x\right)\\ =f_1\left(\varphi \right(x\left)\right)d\eta \left(f_2\right)+d\eta \left(f_1\right)f_2\left(\varphi \right(x\left)\right)\end{array}$

so that $d\eta$ is a tangent vector to $Y$ at $\varphi \left(x\right)$.

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

$0⟶{𝒪}_X\stackrel{d}{⟶}{\Omega }^1\left(X\right)\stackrel{d}{⟶}{\Omega }^2\left(X\right)⟶\cdots$

where the $p$-forms on $X$ are the elements of

${\Omega }^p\left(X\right)={\Lambda }^p\left({\Omega }^1\right(X\left)\right),{\Omega }^1\left(X\right)=I/I^2,\text{where}I=\ker \left(\begin{array}{ccc}{𝒪}_x\otimes {𝒪}_X& ⟶& {𝒪}_X\\ f_1\otimes f_2& \mapsto & f_1{f}_2\end{array}\right)$

and $d$ is the unique anitderivation of degree $1$ extending

$\begin{array}{cccc}d:& {𝒪}_X& ⟶& {\Omega }^1\left(X\right)\\ & f& \mapsto & f\otimes 1–1\otimes f\end{array}$

Then ${𝒪}_X$ acts on ${\Omega }^1\left(X\right)$ by

$f(\sum g_i\otimes f_i)=\sum f{g}_i\otimes h_i=\sum g_i\otimes f{h}_i\text{mod}{I}^2$,

for $f\in {𝒪}_X$ and $\sum g_i\otimes h_i\in I$. As ${𝒪}_X$-modules

$\begin{array}{ccc}{\mathrm{Hom}}_{{𝒪}_X}({\Omega }_X^1,{𝒪}_X)& \stackrel{\sim }{⟶}& \mathrm{Der}\left({𝒪}_X\right)\\ \omega & \mapsto & \omega d\end{array}$

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

$\begin{array}{c}{f}_1\left(\omega d\right)\left(f_2\right)+\left(\omega d\right)\left(f_1\right)f_2=f_1\omega (f_2\otimes 1–1\otimes f_2)+\omega (f_1\otimes 1–1\otimes f_1)f_2\\ =\omega (f_1{f}_2\otimes 1–f_1\otimes f_2)+\omega (f_1\otimes f_2–1\otimes f_1{f}_2)\\ =\omega (f_1{f}_2\otimes 1–1\otimes f_1{f}_2)=\left(\omega d\right)\left(f_1{f}_2\right).\end{array}$

If ${\Omega }^1\left(X\right)$ is a reflexive ${𝒪}_X$-module then

${\Omega }^1\left(X\right)={\mathrm{Hom}}_{{𝒪}_X}\left(\mathrm{Der}\right({𝒪}_X),{𝒪}_X)$.

Example. Let ${𝒪}_X=𝔽[x_1,\dots ,x_n]$ so that $X={𝔽}^n$. If $v\in {𝔽}^n$ then the homomorphism

$\begin{array}{ccc}{𝒪}_X& ⟶& \frac{𝔽\left[t\right]}{\mathrm{langle}{t}^2\mathrm{rangle}}\hfill \\ f& \mapsto & f(x+tv)=f\left(x\right)+t{\partial }_v{\mid }_x\hfill \end{array}$

defines the derivative ${\partial }_v$ in the direction $v$,

${\partial }_v=\sum _{i=1}^n{v}_i\frac{\partial \phantom{m}}{\partial x_i}\text{,}$ if $v=(v_1,\dots ,v_n)$.

Then

• $\left\{\text{sections of}T\right(X\left)\right\}$ has ${𝒪}_X$-basis $\{\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n}\}$ and
• $\left\{\text{sections of}{T}^{*}\right(X\left)\right\}$ has ${𝒪}_X$-basis $\{d{x}_1,\dots ,d{x}_n\}$

and ${\Omega }^p\left(X\right)={𝒪}_X\text{-span}\{d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_p}\mid 1\le i_1<\cdots <i_p\le n\}$ with

$df=\sum _{i=1}^n\frac{\partial f}{\partial x_i}d{x}_i$ 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 $f\in 𝔽[x_1,\dots ,x_n]$.