26. Koszul, Gorenstein, Lci We Were Tacitly Assuming That the Length of a Maximal Regular Sequence Does Not Depend on the Sequence

26. Koszul, Gorenstein, lci We were tacitly assuming that the length of a maximal regular sequence does not depend on the sequence. We would also like to show that if a sequence in a local ring is regular the so is any permutation (this is not true for non-local rings). Theorem 26.1. Suppose R is a local ring with maximal ideal m and a ﬁnitely generated module M. The following are equivalent:

• There is a regular sequence (x1, ..., xn) for M of lenght n • Exti(R/m, M) = 0 for i < n • Any regular sequence for M can be extended to one of length n. (Review Ext as analogous to Tor) i Proof. (1) implies (2): By induction we have Ext (R/m, M/(x1M)) = 0 for i < n − 1. Looking at the long exact sequence of 0 → M → M → M/x1M → 0 shows i that x1 is an injective map on Ext (R/m, M). However it is also the 0 map as it kills R/m, so Exti(R/m, M) = 0 (2) implies (3). Suppose n = 1. Then Hom(R/m, M) = 0 by assumption. So m is not an associated prime of M. The union of the ﬁnite set of associated primes of M is the set of zero divisors of M, so there must be some element of m that is not a zero divisor of M, which is wahat we had to prove. Suppose n > 1. Pick a nonzero divisor x1 of M and apply induction to M/x1M, using the long exact sequence of 0 → M → M → M/x1M → 0. (3) implies (1): Trivial. Corollary 26.2. All maximal regular sequences of M have the same length Corollary 26.3. The quotient of any regular local ring by a regular sequence if Cohen-Macaulay. For example, any hypersurface singularity is Cohen-Macaulay. The Koszul complex. The Koszul complex depends on a sequence of elements (x1, ..., xn) of elements of R. For n = 1 the Koszul complex is just 0 → R →×x1 R

It is exact if x1 is not a zero divisor. For n = 2 the Koszul complex is 0 → R → R2 → R where the ﬁrst arrow takes 1 to (x1, x2) and the second takes (a, b) to x2a − x1b. Note the minus sign. In general the Koszul complex of (x1, ..., xn) is → n n n 0 → R R(1) → R(2) → · · · → R

It is constructed by splicing two copies of the Koszul complex of (x1, ..., xn−1) → (n) (n) n 0 → R R 1 → R 2 → · · · → R → R/(x1, ..., xn−1) → 0 → (n) (n) n 0 → R R 1 → R 2 → · · · → R → R/(x1, ..., xn−1) → 0 with the vertical maps alternately xn and −xn. If the Koszul complex for (x1, ..., xn−1 is exact, then it is easy to check that so is the complex for n, except possibly near 68 RICHARD BORCHERDS the right. However if xn is injective on R/(x1, ..., xn−1) then a little diagram chasing shows that the complex is exact everywhere. (Expand this...) Summary: if (x1, , , xn) is regular, then the Koszul complex is a finite free reso- lution of R/(x1, ..., xn). Using the Koszul complex we can show that regularity of a sequence in a Noe- therian local ring does not depend on its order; in fact in this case it is equivalent to vanishing of the homology of the Koszul complex, which is symmetric in the 0 0 variables xi. For example, if H1(K(x, y)) = 0 then H (K(x) = yH (K(x)) As y is in the maximal ideal and everything is fg we have H0(K(x) = 0 so x is not a zero divisor, so (x, y) is regular. This shows that (x, y) is regular if and only if (y, x) is regular. A similar argument shows that we can exchange the order of any 2 elements of a regular sequence, so regularity does not depend on the order (for Noetherian local rings). Gorenstein rings. These are CM rings with a sort of duality property. A 0- dimensional ring is Gorenstein if HomR(k, R) is 1-dimensional over k = R/m. Informally this says that R is the same ”upside down” meaning that its dual module D(R), where D(M) = HomR(M, k), is isomorphic to R. Examples. k[[x]]/(xn) is Gorenstein. k[[x, y]]/(x2, xy, y2) is not Gorenstein. k[[x, y]]/(x2, y2) is Gorenstein. d In higher dimensions a local ring of dimension d is called Gorenstein if ExtR(k, R) has lenght 1. This definition is due to Grothendieck, who named it after the group- theorist Gorenstein who had proved a result about plane curve singularities equivalent to saying they satisfy this condition. Gorenstein used to tell people he did not understand the definition of a Gorenstein ring. Fortunately in testing whether rings are Gorenstein it is not necessary to know anything about Ext, because of the following result: a Noetherian local ring R of dimension ¿0 is Gorenstein if and only if R/(x) is Gorenstein for a non zero divisor in the maximal ideal. Whether or not a local ring is Gorenstein is quite a subtle property, and seemingly trivial changes can change whether a ring is Gorenstein. Example. Take the group Z/3Z acting on C2. There are two ways it can act without fixed points: the generator may act as either (ω, ω) or as (ω, ω−1) on a basis, where ω is a primitive cube root of unity. The local ring at the origin is Gorenstein in the second example but not the first.