Connections

## de Rham cohomology

Let $A$ be a commutative algebra. The de Rham cohomology of $A$ is the cohomology of the complex

$⋯ ⟶ Ω i – 1 ( A ) ⟶ d i – 1 Ω i ( A ) ⟶ d i Ω i + 1 ( A ) ⟶ ⋯$

where the $p$-differential forms of $A$ is

$Ω p ( A ) = Λ p ( Ω 1 ( A ) ) , Ω 1 ( A ) = I / I 2 , I = ker ( A ⊗ A → A ) ,$

and $d$ is the unique antiderivation of degree 1 which extends

$d : A ⟶ Ω 1 ( A ) x ⟼ x ⊗ 1 – 1 ⊗ x and satisfies d 2 = 0$.

Let $M$ be an $A$-module. A connection on $M$ is an $𝔽$-linear map

$∇ : M ⟶ M ⊗ A Ω 1 ( A )$such that ( f m ) = f ( m ) + m d f ,

for $f\in A$, $m\in M$. There is a unique extension of $\nabla$ to

$⋯ ⟶ M ⊗ A Ω i – 1 ( A ) ⟶ ∇ M ⊗ A Ω i ( A ) ⟶ ∇ M ⊗ A Ω i + 1 ( A ) ⟶ ⋯$

such that

The curvature of $\nabla$ is

$R : M ⟶ M ⊗ A Ω 2 ( A ) given by R = ∇ 1 ∘ ∇ 0 ,$

and $\nabla$ is flat if $R=0$.

If $\nabla$ is a flat connection then (??) is a complex and the de Rham cohomology of $\left(M,\nabla \right)$ is the homology of (??).

Then $A$ acts on ${\Omega }^{1}\left(A\right)$ by

$f ( ∑ g i ⊗ f i ) = ∑ f g i ⊗ h i = ∑ g i ⊗ f h i mod I 2$,

for $f\in A$ and $\sum {g}_{i}\otimes {h}_{i}\in I$. As $A$-modules

$Hom A ( Ω 1 ( A ) , A ) ⟶ ∼ Der ( A ) φ ⟼ φ d$

and, if ${\Omega }^{1}\left(A\right)$ is a reflexive $A$-module then

$Ω 1 ( A ) = Hom A ( Der ( A ) , A )$.

Let $\nabla$ be a connection on $M$ and define

$Der ( A ) ⟶ End 𝔽 ( M ) ∂ ⟼ ∇ ∂ by ∇ ∂ = ( part ⊗ id M ) ∘ ∇$,

so that ${\nabla }_{\partial }:M\stackrel{\nabla }{\to }{\Omega }^{1}\left(A\right){\otimes }_{A}M\stackrel{\partial \otimes {\mathrm{id}}_{M}}{\to }A{\otimes }_{A}M=M$. Then, for $f,{f}_{1},{f}_{2}\in A$, $\partial ,{\partial }_{1},{\partial }_{2}\in \mathrm{Der}\left(A\right)$ and $m\in M$,

$∇ ∂ ( f m ) = ∂ ( f ) m + f ∇ ∂ ( m ) , and ∇ f 1 ∂ 1 + f 2 ∂ 2 ( m ) = f 1 ∇ ∂ 1 ( m ) + f 2 ∇ ∂ 2 ( m ) .$

If ${\Omega }^{1}\left(A\right)$ is a reflexive $A$-module then the connection $\nabla$ is determined by the map $\partial \to {\nabla }_{\partial }$ with the properties (a) and (b).

## Derivations

Let $A$ be a ring and let $M$ be an $\left(A,A\right)$-bimodule. A derivation is an $\mathrm{Fopf}$-linear map $\mathrm{part}?A?M$ such that

Let

$I = I = ker ( A ⊗ A ⟶ A f 1 ⊗ f 2 ⟼ f 1 f 2 ) and d : A ⟶ I f ⟼ f ⊗ 1 – 1 ⊗ f .$

Then $d:A\to I$ is a derivation and if $\partial :A\to M$ is a derivation then there exists a unique $\left(A,A\right)$-module map $\varphi :I\to M$ such that $\partial =\varphi \circ d$.

$A ⟶ ∂ M d ↓ ↗ ϕ I In other words, Der ( A , M ) = Hom A ⊗ A ( I , M )$.

If $A$ is commutative and $M$ is an $A$-module then $M$ is an $\left(A,A\right)$-bimodule on which $I$ acts by $0$ (since $\left(f\otimes 1–1\otimes f\right)m=\mathrm{fm}–\mathrm{mf}=0$). In fact, since the multiplication map $A\otimes A\to A$ is surjective, $A\otimes A\otimes A\right)/I$ and $A$-modules are the same thing as $A\otimes A$-modules on which $I$ acts by 0, and (*) becomes

$Der ( A , M ) = Hom A ⊗ A ( I , M ) = Hom A ( I / I 2 , M )$.

Thus, $\mathrm{Der}\left(A\right)$ is dual to the space ${\Omega }^{1}=I/{I}^{2}$ of 1-forms for $A$,

$Der ( A ) = Hom A ( Ω 1 , A )$.