We are now going to discuss another mechanism for determining the length of maximal -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 be a finitely generated -module. Consider the graded commutative algebra with the product given by the wedge product; the graded commutativity is similar to the cup-product in cohomology, and implies that

Given , we can define a *differential* on as follows. Namely, we define

(More precisely, this clearly defines an alternating map , and this thus factors through the alternating product by the universal property.) It is very easy to see that . Moreover, is an anti-derivation. If are homogeneous elements of the graded algebra, then

Definition 20The complex, together with the multiplicative structure, just defined is called theKoszul complexand is denoted .

The special case we shall care the most about is when and is given by the dot product with a vector . Then we shall write

for the Koszul complex. Now that we have a complex, we can define its homology (and cohomology).

Definition 21Let be an -module. We write for the complex (and similarly when is ). The homology of this complex is called theKoszul homologyof and is denoted (or ).

One of the basic facts about this is that is an exact functor if is flat. In particular, is an exact functor as each of the terms of the complex are free. In particular,

Theorem 22If is a flat module, then is a -functor.

We can also dualize everything.

Definition 23Let be an -module. We write for the complex . The cohomology of this cochain complex is called theKoszul cohomologywith coefficients and is denoted . (Similarly we define when is a free module, and write for the Koszul cohomology.)

It is similarly easy to see that when is projective, then will be an exact functor in , and the Koszul cohomology will be a -functor where the connecting homomorphisms raise the degree.

So what can we do with this? Well, as we will see the Koszul complex detects the regularity of sequences. This is a nontrivial fact, and it will basically rely on the fact that, first of all, the Koszul complex for (i.e. the complex of the free module together with the functional ) is very simple; it’s

It will rely on this simple observation and the fact, proved next, that Koszul complexes behave nicely with respect to tensoring. Given graded -algebras , we can form the *graded tensor product* . By definition, this is just as a graded module, but the multiplicative structure is slightly different. Namely, we define the product of as

if the elements in question are homogeneous. This is a fairly common construction. When one has finite-dimensional CW complexes , for instance, the cohomology ring of with coefficients in a field is the graded tensor product of the cohomology rings of and . Another example, more relevant here, is that if are two modules, then the exterior algebra is the graded tensor product of and .

Proposition 24Let be linear functionals. Then the Koszul complex is the tensor product 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, Here is the functional where the last map is addition. So for instance this implies that for two tuples . This implies that in the case we care about most, catenation of lists of elements corresponds to the tensor product.

I think I’m going to prove this result tomorrow. It’s all right; we don’t have to rush.

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