## Koszul and de Rham complexes

Last update: 22 June 2012

## Example 1: Singular homology

Let ${e}_{0},\dots ,{e}_{N–1}$ be the standard basis of $ℝ$. The standard $n$-simplex is $Δn= {x0e0 +⋯+ xnen∣ x0+⋯+ xn≤1}, with faces defined by ιj: Δn-1 ⟶Δn$ where ${\iota }_{j}\left({x}_{0}{e}_{0}+\cdots +{x}_{n-1}{e}_{n-1}\right)={x}_{0}{e}_{0}+\cdots +{x}_{j-1}{e}_{j-1}+{x}_{j}{e}_{j}+{x}_{j}{e}_{j+1}+\cdots +{x}_{n-1}{e}_{n}$.

Let $X$ be a topological space and let $𝔸$ be a ring. The singular homology of $X$ is the homology ${H}_{i}\left(X;𝔸\right)$ of the complex with given by $d(ef) = ∑j=1n (-1)j ef∘ij$ If $Y$ is a subspace of $X$ let $C\left(X;Y;𝔸\right)=C\left(X;𝔸\right)/C\left(Y;𝔸\right)$ so that $0⟶ C(Y;𝔸)⟶ C(X;𝔸)⟶ C(X;Y;𝔸) ⟶0$ is an exact sequence of complexes.

## Example 2: Cell complex homology

Let ${B}_{n}$ be the $n$-dimensional open ball in ${ℝ}^{n}$.

Let $X$ be a Hausdorff topological space. A cellular decomposition of $X$ is a sequence $∅⊆X0⊆ X1⊆⋯⊆ XN=X$ of closed subspaces of $X$ such that, for each $n$, ${X}_{n}-{X}_{n-1}$ has a finite number of connected components and, for each connected component $C$ of ${X}_{n}-{X}_{n-1}$ there is a homeomorphism ${B}_{n}\stackrel{\sim }{⟶}C$ which extends to a continuous map $\stackrel{‾}{{B}_{n}}⟶X$.

The cellular homology is the homology of the complex $⋯⟶Γn ⟶dn Γn-1 ⟶⋯$ given by ${\Gamma }_{n}={H}_{n}\left({X}_{n–1},{X}_{n}\right)$ with the connecting homomorphism ${d}_{n}:{H}_{n}\left({X}_{n},{X}_{n-1}\right)⟶{H}_{n-1}\left({X}_{n-1},{X}_{n-2}\right)$ coming from the exact sequence $0⟶ C(Xn-1, Xn-2) ⟶C(Xn, Xn-2) ⟶C(Xn, Xn-1) ⟶0.$ Then the singular homology of $X$ is isomorphic to the cellular homology, $Hn(X) ≃Hn(Γ).$

## The Koszul complex and the de Rham complex

Let $𝔸$ be a commutative ring, $L$ an $𝔸$-module and $u:L⟶𝔸$ an $𝔸$-linear map. The Koszul complex is $⋯⟶ Λi+1 (L)⟶ di+1 Λi(L)⟶ di Λi-1 (L)⟶⋯$ where $d$ is the unique antiderivation ($d\left(x\wedge y\right)=d\left(x\right)\wedge y-x\wedge d\left(y\right)$) of $\Lambda \left(L\right)$ that extends $u:L\to 𝔸$. Explicitly, $d(ℓ1∧⋯ ∧ℓn) = ∑i=1n (-1)i+1 u(ℓi) , ℓ1∧⋯∧ ℓi-1 ∧ℓi+1 ∧⋯∧ℓn.$

#### Example

Let $𝕂$ be a commutative ring, $L$ a $𝕂$-module and let $𝔸=S\left(L\right)$. The Koszul complex for the $S\left(L\right)$-module $S\left(L\right){\otimes }_{𝕂}L$ with linear form $u: S(L)⊗𝕂 L ⟶ S(L) f⊗x ↦ fx is Λ(S(L) ⊗𝕂L) =S(L)⊗ 𝕂Λ(L)$ and is the direct sum of complexes $0⟶S0L ⊗𝕂Λn L⟶S1L ⊗𝕂 Λn-1 ⟶⋯⟶ SnL⊗ 𝕂Λ0L ⟶0$ over $n\in {ℤ}_{\ge 0}$ with $d((x1⋯ xp)⊗( y1∧⋯∧ yq)) =∑i=1 q (-1) i+1yi x1⋯xp ⊗(y1 ∧⋯∧ yi-1∧ yi+1∧ ⋯∧yq).$ If $L$ is flat or $A$ is a $ℚ$-algebra these complexes are exact and $∑i=1n (-1)i [Si(L)] [Λn-i (L)]=0.$

#### Example 2

Let $𝕂$ be a commutative ring, $M$ a $𝕂$-module and ${x}_{1},\dots ,{x}_{\ell }$ a set of commuting endomorphisms of $M$. Then $M$ is a module for the ring $𝔸 =𝕂[x1, …,xℓ] and if L=𝔸⊕ℓ =𝔸-span {e1,…, eℓ}$ with linear map $u: L ⟶ 𝔸 ei ↦ Xi$ If ${C}^{p}\left(M\right)=\left\{\phantom{\rule{0.5em}{0ex}}\text{alternating maps from}\phantom{\rule{0.5em}{0ex}}\left\{1,\dots ,\ell {\right\}}^{p}⟶M\right\}$ then $Cp(M)≃ Hom𝔸(Λp (L),M)≃ Hom𝕂(Λp (𝕂ℓ),M) and Cp(M)≃ M⊗𝕂Λp (𝕂ℓ)$ and this gives a double complex $⋯⇋Cp-1 (M)⇋∂p ∂p-1 Cp(M) ⇋∂p+1 ∂p Cp+1 (M)⇋⋯$ with $(∂pm) (α1,…, αp+1) =∑j=1 p+1(-1) j+1x αjm( α1,…, αj-1, αj+1, …, αp+1)$ and $\left({\partial }^{p}m\right)\left({\alpha }_{1},\dots ,{\alpha }_{p+1}\right)=$

The homology and cohomology of the complex are denoted ${H}_{r}\left({x}_{1},\dots ,{x}_{\ell };M\right)$ and ${H}^{r}\left({x}_{1},\dots ,{x}_{\ell };M\right)$.

Let $𝔸$ be a commutative ring and let $M$ be an $𝔸$-module. A sequence ${x}_{1},\dots ,{x}_{\ell }$ of elements of $𝔸$ is completely secant for $M$ if ${H}_{r}\left({x}_{1},\dots ,{x}_{\ell };M\right)=0$, for $i\in {ℤ}_{>0}$. An $M$-regular sequence is a sequence ${x}_{1},\dots ,{x}_{\ell }$ of elements of $𝔸$ such that $M (x1 M+⋯+ xi–1M) ⟶ M (x1M+⋯ +xi-1 M) y ↦ xiy is injective for i=1,2,…,n .$

If ${x}_{1},\dots ,{x}_{\ell }$ is an $M$-regular sequence then ${x}_{1},\dots ,{x}_{\ell }$ is completely secant for $M$. If ${x}_{1},\dots ,{x}_{\ell }$ is an $𝔸$-regular sequence and $I=⟨{x}_{1},\dots ,{x}_{\ell }⟩$ then $I/{I}^{2}$ is free of rank $r$ over $𝔸/I$ (see [Lang, XXI Sec. 4]).

Example. The special case $M=𝕂\left[{x}_{1},\dots ,{x}_{n}\right]$ with commuting endomorphisms $\frac{\partial \phantom{x}}{\partial {x}_{1}},\dots ,\frac{\partial \phantom{x}}{\partial {x}_{n}}$ is the de Rham complex of $𝕂\left[{x}_{1},\dots ,{x}_{n}\right]$.

### de Rham cohomology

Let $A$ be a commutative algebra. The de Rham cohomology of $A$ is the cohomology of the complex $⋯⟶ Ωi-1(A) ⟶di-1 Ωi(A)⟶ di Ωi+1 (A)⟶⋯$ where the $p$-differential forms of $A$ is ${\Omega }^{p}\left(A\right)={\Lambda }^{p}\left({\Omega }^{1}\left(A\right)\right),{\Omega }^{1}\left(A\right)=I/{I}^{2},I=\mathrm{ker}\left(A\otimes A\to A\right),$ and $d$ is the unique antiderivation of degree 1 which extends $d: A ⟶ Ω1(A) x ↦ x⊗1-1⊗x$ and satisfies ${d}^{2}=0$.

Example. If $A=𝔽\left[{x}_{1},\dots ,{x}_{n}\right]$ then ??????

Let $M$ be an $A$-module. A connection on $M$ is an $𝔽$-linear map $\nabla :M⟶M{\otimes }_{A}{\Omega }^{1}\left(A\right)$ such that $\nabla \left(fm\right)=f\nabla \left(m\right)+m\otimes df$, for $f\in A$, $m\in M$.

## Notes and References

These notes are originally from http://researchers.ms.unimelb.edu.au/~aram@unimelb/MathGlossary/index.html the file http://researchers.ms.unimelb.edu.au/~aram@unimelb/MathGlossary/Koszul_deRham.xml

## References

[BouA] N. Bourbaki, Algebra I, Chapters 1-3, Elements of Mathematics, Springer-Verlag, Berlin, 1990.

[BouL] N. Bourbaki, Groupes et Algèbres de Lie, Chapitre IV, V, VI, Eléments de Mathématique, Hermann, Paris, 1968.