Koszul and de Rham Complexes

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

Example 1: Singular homology

Let $e_0,\dots ,e_{N–1}$ be the standard basis of $ℝ$. The standard $n$-simplex is

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

given by

If $Y$ is a subspace of $X$ let $C(X;Y;𝔸)=C(X;𝔸)/C(Y;𝔸)$ so that

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 $d_n:H_n(X_n,X_{n–1})\mathrm{xrarr}{H}_{n–1}(X_{n–1},X_{n–2})$ 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,

an $𝔸$-linear map. The Koszul complex is

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,

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

and is the direct sum of complexes

over $n\in {\mathbb{Z}}_{\ge 0}$ with

If $L$ is flat or $A$ is a $\mathbb{Q}$-algebra these complexes are exact and

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

with linear map

If $C^p\left(M\right)=\left\{\text{alternating maps from}\right\{1,\dots ,\ell {\}}^p⟶M\}$ then

and this gives a double complex

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

$$\begin{array}{ccc}\frac{M}{(x_{1}M+\cdots +x_{i–1}M)}& ⟶& \frac{M}{(x_{1}M+\cdots +x_{i–1}M)}\\ y& \mapsto & x_{i}y\end{array}$$ 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 $I=忯x_1,\dots ,x_{\ell }濯$ 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

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 $A=𝔽[x_1,\dots ,x_n]\mathrm{then}$

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 $f\in A$, $m\in M$.