Koszul and de Rham complexes

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

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 $${\Delta }_n=\{x_0{e}_0+\cdots +x_n{e}_n\mid x_0+\cdots +x_n\le 1\},\text{with faces defined by}{\iota }_j:{\Delta }_{n-1}⟶{\Delta }_n$$ where ${\iota }_j(x_0{e}_0+\cdots +x_{n-1}{e}_{n-1})=x_0{e}_0+\cdots +x_{j-1}{e}_{j-1}+x_j{e}_j+x_j{e}_{j+1}+\cdots +x_{n-1}{e}_n$.

Let $X$ be a topological space and let $𝔸$ be a ring. The singular homology of $X$ is the homology $H_i(X;𝔸)$ of the complex with $$C_n(X;𝔸)=𝔸\text{-span}\left\{e_f\right|f:{\Delta }_n⟶X\text{ is continuous}\}\text{and}d:C_n(X;𝔸)⟶C_{n-1}(X;𝔸)$$ given by $$d\left(e_f\right)=\sum _{j=1}^n{(-1)}^j{e}_{f\circ i_j}$$ If $Y$ is a subspace of $X$ let $C(X;Y;𝔸)=C(X;𝔸)/C(Y;𝔸)$ 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 $\varnothing \subseteq X_0\subseteq X_1\subseteq \cdots \subseteq X_N=X$ 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 $\overline{B_n}⟶X$.

The cellular homology is the homology of the complex $$\cdots ⟶{\Gamma }_n\stackrel{d_n}{⟶}{\Gamma }_{n-1}⟶\cdots$$ given by ${\Gamma }_n=H_n(X_{n–1},X_n)$ with the connecting homomorphism $d_n:H_n(X_n,X_{n-1})⟶H_{n-1}(X_{n-1},X_{n-2})$ coming from the exact sequence $$0⟶C(X_{n-1},X_{n-2})⟶C(X_n,X_{n-2})⟶C(X_n,X_{n-1})⟶0.$$ Then the singular homology of $X$ is isomorphic to the cellular homology, $$H_n\left(X\right)\simeq H_n\left(\Gamma \right).$$

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 $$\cdots ⟶{\Lambda }^{i+1}\left(L\right)\stackrel{d_{i+1}}{⟶}{\Lambda }^i\left(L\right)\stackrel{d_i}{⟶}{\Lambda }^{i-1}\left(L\right)⟶\cdots$$ where $d$ is the unique antiderivation ($d(x\wedge y)=d\left(x\right)\wedge y-x\wedge d\left(y\right)$) of $\Lambda \left(L\right)$ that extends $u:L\to 𝔸$. Explicitly, $$d({\ell }_1\wedge \cdots \wedge {\ell }_n)=\sum _{i=1}^n{\left(-1\right)}^{i+1}u\left({\ell }_i\right)\phantom{,}{\ell }_1\wedge \cdots \wedge {\ell }_{i-1}\wedge {\ell }_{i+1}\wedge \cdots \wedge {\ell }_n.$$

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 $$\begin{array}{cccc}u:& S\left(L\right){\otimes }_{𝕂}L& ⟶& S\left(L\right)\\ & f\otimes x& \mapsto & fx\end{array}\text{is}\Lambda \left(S\right(L\left){\otimes }_{𝕂}L\right)=S\left(L\right){\otimes }_{𝕂}\Lambda \left(L\right)$$ and is the direct sum of complexes $$0⟶S^{0}L{\otimes }_{𝕂}{\Lambda }^{n}L⟶S^{1}L{\otimes }_{𝕂}{\Lambda }^{n-1}⟶\cdots ⟶S^{n}L{\otimes }_{𝕂}{\Lambda }^{0}L⟶0$$ over $n\in {\mathbb{Z}}_{\ge 0}$ with $$d\left(\right(x_1\cdots x_p\left)\otimes \right(y_1\wedge \cdots \wedge y_q\left)\right)=\sum _{i=1}^q{\left(-1\right)}^{i+1}{y}_i{x}_1\cdots x_p\otimes (y_1\wedge \cdots \wedge y_{i-1}\wedge y_{i+1}\wedge \cdots \wedge y_q).$$ If $L$ is flat or $A$ is a $\mathbb{Q}$-algebra these complexes are exact and $$\sum _{i=1}^n{\left(-1\right)}^i\left[S^i\right(L\left)\right]\left[{\Lambda }^{n-i}\right(L\left)\right]=0.$$

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 $$𝔸=𝕂[x_1,\dots ,x_{\ell }]\text{and if}L={𝔸}^{\oplus \ell }=𝔸\text{-span}\{e_1,\dots ,e_{\ell }\}$$ with linear map $$\begin{array}{cccc}u:& L& ⟶& 𝔸\\ & e_i& \mapsto & X_i\end{array}$$ If $C^p\left(M\right)=\left\{\text{alternating maps from}\right\{1,\dots ,\ell {\}}^p⟶M\}$ then $$C^p\left(M\right)\simeq {\mathrm{Hom}}_{𝔸}\left({\Lambda }^p\right(L),M)\simeq {\mathrm{Hom}}_{𝕂}\left({\Lambda }^p\right({𝕂}^{\ell }),M)\text{and}{C}^p\left(M\right)\simeq M{\otimes }_{𝕂}{\Lambda }^p\left({𝕂}^{\ell }\right)$$ and this gives a double complex $$\cdots \leftrightharpoons C^{p-1}\left(M\right)\underset{{\partial }_p}{\overset{{\partial }^{p-1}}{\leftrightharpoons }}{C}^p\left(M\right)\underset{{\partial }_{p+1}}{\overset{{\partial }^p}{\leftrightharpoons }}{C}^{p+1}\left(M\right)\leftrightharpoons \cdots$$ with $$\left({\partial }^{p}m\right)({\alpha }_1,\dots ,{\alpha }_{p+1})=\sum _{j=1}^{p+1}(-1{)}^{j+1}{x}_{{\alpha }_j}m({\alpha }_1,\dots ,{\alpha }_{j-1},{\alpha }_{j+1},\dots ,{\alpha }_{p+1})$$ and $\left({\partial }^{p}m\right)({\alpha }_1,\dots ,{\alpha }_{p+1})=$

The homology and cohomology of the complex are denoted $H_r(x_1,\dots ,x_{\ell };M)$ and $H^r(x_1,\dots ,x_{\ell };M)$.

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 $H_r(x_1,\dots ,x_{\ell };M)=0$, for $i\in {\mathbb{Z}}_{>0}$. An $M$-regular sequence is a sequence $x_1,\dots ,x_{\ell }$ of elements of $𝔸$ such that $$\begin{array}{ccc}\frac{M}{(x_{1}M+\cdots +x_{i–1}M)}& ⟶& \frac{M}{(x_{1}M+\cdots +x_{i-1}M)}\\ y& \mapsto & x_{i}y\end{array}\text{is injective for}i=\mathrm{1,2},\dots ,n.$$

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 $M=𝕂[x_1,\dots ,x_n]$ with commuting endomorphisms $\frac{\partial \phantom{x}}{\partial x_1},\dots ,\frac{\partial \phantom{x}}{\partial x_n}$ is the de Rham complex of $𝕂[x_1,\dots ,x_n]$.

de Rham cohomology

Let $A$ be a commutative algebra. The de Rham cohomology of $A$ is the cohomology of the complex $$\cdots ⟶{\Omega }^{i-1}\left(A\right)\stackrel{d_{i-1}}{⟶}{\Omega }^i\left(A\right)\stackrel{d_i}{⟶}{\Omega }^{i+1}\left(A\right)⟶\cdots$$ where the $p$-differential forms of $A$ is ${\Omega }^p\left(A\right)={\Lambda }^p\left({\Omega }^1\right(A\left)\right),{\Omega }^1\left(A\right)=I/I^2,I=\ker (A\otimes A\to A),$ and $d$ is the unique antiderivation of degree 1 which extends $$\begin{array}{cccc}d:& A& ⟶& {\Omega }^1\left(A\right)\\ & x& \mapsto & x\otimes 1-1\otimes x\end{array}$$ and satisfies $d^2=0$.

Example. If $A=𝔽[x_1,\dots ,x_n]$ 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.

page history