I’ve not been a very good MaBloWriMo participant this time around. Nonetheless, coursework does tend to sap the time and energy I have for blogging. I have been independently looking as of late at the formal function theorem in algebraic geometry, which can be phrased loosely by saying that the higher direct images under a proper morphism of schemes commute with formal completions. This is proved in Hartshorne for projective morphisms by first verifying it for the standard line bundles and then using a (subtle) exactness argument, but EGA III.4 presents an argument for general proper morphisms. The result is quite powerful, with applications for instance to Zariski’s main theorem (or at least a weak version thereof), and I would like to say a few words about it at some point, at least after I have a fuller understanding of it than I do now. So I confess to having been distracted by algebraic geometry.

For today, I shall continue with the story on the Koszul complex, and barely begin the connection between Koszul homology and regular sequences. Last time, we were trying to prove:

Proposition 24 Let ${\lambda: L \rightarrow R, \lambda': L' \rightarrow R}$ be linear functionals. Then the Koszul complex ${K_*(\lambda \oplus \lambda')}$ is the tensor product ${K_*(\lambda) \otimes K_*(\lambda')}$ as differential graded algebras.

So in other words, not only is the algebra structure preserved by taking the tensor product, but when you think of them as chain complexes, ${K_*(\lambda \oplus \lambda') \simeq K_*(\lambda) \oplus K_*(\lambda')}$. This is a condition on the differentials. Here ${\lambda \oplus \lambda'}$ is the functional ${L \oplus L' \stackrel{\lambda \oplus \lambda'}{\rightarrow} R \oplus R \rightarrow R}$ where the last map is addition.

So for instance this implies that ${K_*(\mathbf{f}) \otimes K_*(\mathbf{f}') \simeq K_*(\mathbf{f}, \mathbf{f}')}$ for two tuples ${\mathbf{f} = (f_1, \dots, f_i), \mathbf{f}' = (f'_1, \dots, f'_j)}$. This implies that in the case we care about most, catenation of lists of elements corresponds to the tensor product.

Before starting the proof, let us talk about differential graded algebras. This is not really necessary, but the Koszul complex is a special case of a differential graded algebra.

Definition 25 A differential graded algebra is a graded unital associative algebra ${A}$ together with a derivation ${d: A \rightarrow A}$ of degree one (i.e. increasing the degree by one). This derivation is required to satisfy a graded version of the usual Leibnitz rule: ${d(ab) = (da)b + (-1)^{\mathrm{deg} a} a (db) }$. Moreover, ${A}$ is required to be a complex: ${d^2=0}$. So the derivation is a differential.

So the basic example to keep in mind here is the case of the Koszul complex. This is an algebra (it’s the exterior algebra). The derivation ${d}$ was immediately checked to be a differential. There is apparently a category-theoretic interpretation of DGAs, but I have not studied this.

Proof: As already stated, the graded algebra structures on ${K_*(\lambda), K_*(\lambda')}$ are the same. This is, I suppose, a piece of linear algebra, about exterior products, and I won’t prove it here. The point is that the differentials coincide. The differential on ${K_*(\lambda \oplus \lambda')}$ is given by extending the homomorphism ${L \oplus L' \stackrel{\lambda \oplus \lambda'}{\rightarrow} R}$ to a derivation. This extension is unique. (more…)

We are now going to discuss another mechanism for determining the length of maximal ${M}$-sequences, namely the Koszul complex. This is going to be super-useful in a whole bunch of ways. For one thing, it is integral in the proof that regular local rings are of finite global dimension, because the Koszul complex becomes a free resolution of the residue field.

Another one, which has excited me as of late, is that if you have a suitable scheme (say, quasi-compact and quasi-separated) and a quasi-coherent sheaf on it, then its Cech cohomology is in fact a direct limit of Koszul cohomologies! So properties of Koszul cohomology can be used to compute the cohomology of projective space as in Hartshorne, and thus to prove the fundamental theorem that higher direct images by projective (and, eventually, proper) morphisms preserve coherence. But this is getting rather far afield of what I want to talk about today.

Let ${L}$ be a finitely generated ${R}$-module. Consider the graded commutative algebra ${K = \bigwedge L = \bigoplus \wedge^i L}$ with the product given by the wedge product; the graded commutativity is similar to the cup-product in cohomology, and implies that

$\displaystyle x \wedge y = (-1)^{\deg x \deg y} y \wedge x.$

Given ${\lambda: L \rightarrow R}$, we can define a differential on ${K}$ as follows. Namely, we define

$\displaystyle d( x_1 \wedge \dots \wedge x_n) = \sum_i (-1)^i\lambda(x_i) x_1 \wedge \dots \wedge \hat{x_i} \wedge \dots \wedge x_n.$

(More precisely, this clearly defines an alternating map ${L^n \rightarrow \wedge^{n-1} L}$, and this thus factors through the alternating product by the universal property.) It is very easy to see that ${d \circ d = 0}$. Moreover, ${d}$ is an anti-derivation. If ${x,y \in K}$ are homogeneous elements of the graded algebra, then

$\displaystyle d(x\wedge y) = d(x)\wedge y + (-1)^x x \wedge d(y)$

Definition 20 The complex, together with the multiplicative structure, just defined is called the Koszul complex and is denoted ${K_*(\lambda)}$.