MATH 250B: COMMUTATIVE ALGEBRA 67 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 equiv- alent 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 (!; !) or as (!; !−1) on a basis, where ! is a primitive cube root of unity. The local ring at the origin is Gorenstein in the second example but not the first..