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 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, . This is a condition on the differentials. 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.
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 together with a derivation of degree one (i.e. increasing the degree by one). This derivation is required to satisfy a graded version of the usual Leibnitz rule: . Moreover, is required to be a complex: . 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 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 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 is given by extending the homomorphism to a derivation. This extension is unique. (more…)