Koszul complexes and local algebra

Let $R$ be commutative and choose elements $f_1,\ldots,f_r$. Take the exterior algebra on a free module with basis $e_1,\ldots,e_r$, putting exterior degree $i$ in chain degree $i$. Define the differential by sending $e_j$ to $f_j$ and extending as a graded derivation of degree minus one. Each pair of removed generators occurs twice in the square of the differential with opposite signs; commutativity of the coefficients makes those terms cancel.

In degree zero, the differential image is exactly the ideal generated by the chosen elements, so zeroth homology is always $R/(f_1,\ldots,f_r)$. That conclusion does not need regularity. Positive homology measures relations not explained by the exterior relations already built into the complex. For two generators, the obvious relation is $f_1(-f_2)+f_2f_1=0$. Exactness asks whether every relation is a multiple of this one and whether that generator of relations has any annihilator.

A regular sequence here means that its generated ideal is proper and each element acts injectively on the quotient by the preceding elements. The order is part of this definition. The theorem states that a regular sequence in a commutative ring has zero positive Koszul homology, so its Koszul complex is a finite free resolution of the quotient. No Noetherian hypothesis is needed for this implication. The proper-ideal condition keeps the quotient nonzero and excludes trivial unit-generated cases from the definition.

One proof proceeds inductively: adjoining the next generator forms, up to the appropriate chain grading, the cone of multiplication by that generator on the preceding Koszul complex. The induction hypothesis leaves only the preceding quotient in degree zero. Injectivity of multiplication on that quotient kills the possible new positive homology, while its cokernel gives the next quotient. This is a proof sketch explaining the role of each regularity condition, rather than an appeal to the exterior construction alone.

For a one-element list, positive Koszul homology is simply the annihilator of that element. Thus a nonzero zero divisor gives an immediate failure of the resolution property. Longer lists require testing injectivity on successive quotients, not only on the original ring. Two elements may each be nonzero divisors in a domain but the second can become a zero divisor after quotienting by the first. The regular-sequence condition is designed to inspect precisely this accumulated change of ring.

In Noetherian local algebra, with a finite module and generators in the maximal ideal, vanishing criteria and minimal resolutions relate Koszul homology to depth and regularity. Those stronger converses need their own hypotheses and are not asserted for arbitrary rings and lists here. Even the forward theorem has substantial computational consequences: tensoring its free resolution with a residue module computes Tor in finitely many degrees. A failed Koszul resolution instead directs attention toward additional syzygies, sometimes leading to the infinite periodic behavior seen earlier.