So last time, when (partially) computing the cohomology of affine space, we used a fact about the Koszul complex. Namely, I claimed that the Koszul complex is acyclic when the elements in question generate the unit ideal. This was swept under the rug, and logically I should have covered that before getting to yesterday’s bit of algebraic geometry. So today, I will backtrack into the elementary properties of the Koszul complex, and prove a more general claim.

** 0.9. A chain-homotopy on the Koszul complex **

Before proceeding, we need to invoke a basic fact about the Koszul complex. If is a Koszul complex, then multiplication by anything in is chain-homotopic to zero. In particular, if generates the unit ideal, then is homotopically trivial, thus exact. This is one reason we should restrict our definition of “regular sequence” (as we do) to sequences that do not generate the unit ideal, or the connection with the exactness of the Koszul complex wouldn’t work as well.

**Proposition 33** * Let . Then the multiplication by map is chain-homotopic to zero. *

*Proof:* Let and let . Then there is a vector . We can define a map of degree one

which is the *interior product* with , i.e. sending to . Now, however, we know that the differential, which we’ll call , is a *derivation* on the Koszul algebra . In particular,

as has degree one. But , from its definition, is just times the unit of the Koszul algebra and . In particular, we find

This implies that multiplication by is homotopically trivial.

As an example, let be a local ring. Let generate . Then any element acts in a way that is homotopically trivial on . In particular, the homology groups are vector spaces (finite dimensional!) over the residue field . From this, we can easily prove:

**Corollary 34** * If is any -module, and , then acts by zero on and . In particular, if , then the Koszul homology and cohomology vanish identically for any module. *

This is now clear, because the chain-homotopy on the Koszul complex passes to a chain homotopy on the Koszul complex with coefficients or the dual Koszul complex.

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