Lecture 25. Conclude the Hilbert Syzygies theorem by -> mention regular local rings, -> extending the theorem to regular local rings, -> the Auslander-Buchsbaum strengthening of the result. I give a quick and amateurish discussion of this. You can find higher powered versions in [Ma] and [Ei]. Direction for the last 5 lectures. Memorable point: Write S = = k[x1,..xN]. A quotient ring A = S/I of dimension d is Cohen-Macaulay <=> depth d. Auslander-Buchsbaum says that A has a free resolution of length <= codim c = N-d Moreover A is Gorenstein <=> Cohen-Macaulay plus the free resolution ends with a free module of rank 1 S(-k). A couple of examples. codim 1. I an ideal in A = k[x1,..xn] with dim A/I = n-1. if A/I is Cohen-Macaulay, then I = (f), so its resolution is of length 1, so is A <-f- A <- 0 On the other hand, A/I may have embedded primes, so saying dim A/I = n-1 without Cohen-Macaulay doesn't given anything -- if A/I has a 0-dimensional associated prime (so A/I has a submodule isomorphic to A/m) then it can only have a resolution of length n. Hilbert-Burch Suppose that a quotient ring A = k[x1,..xn]/I (1) has codimension 2, and (2) has a free resolution of length 2 Then the free resolution is necessarily of the form A <-F- (d+1)*A <-M- d*A <- 0 | v A/I where M is a (d+1) x d matrix and F is the vector \Wedge^d M made up by its d x d minors (in appropriate order and with appropriate signs). The proof is also memorable: the module we are resolving has rank 1, and the sequence is exact outside V(I). In particular it is exact after tensoring with the function field k(x1.. xn). Therefore the ranks are d+1 and d (so that the alternating sum of 1,d+1,d comes to 0). Next, still arguing over k(x1.. xn), the cokernel of a (d+1) x d matrix of rank d is given by its dxd minors (Cramer's rule) up to proportionality. If the constant of proportionality is not a unit, then A/I has a associate prime of dimension n-1, which contradicts the assumption that dim A/I = n-2. ==== Cohen-Macaulay and Gorenstein rings Preliminary section introducing Ext_A^r(M,N) Q. Why the name Ext? Ext is about the Hom functor Hom_A(M,?) is a covariant left exact functor, as we have seen: If 0 -> N1 -> N2 -> N3 -> 0 is a ses then 0 -> Hom_A(M,N1) -i-> Hom_A(M,N2) -p-> Hom_A(M,N3) is exact. The cokernel of p at the right end measures _extensions_ of M by N1. I describe this briefly. Given the ses and a homomorphism f: M -> N3, consider the set theoretic fibre product Ef in N2 + M defined by Ef = { n,m | p(n) = f(m) }. E has projections into the two summands, and fits into a cartesian square. One sees that the kernel of E -> M is a copy of N1 in E fitting into an extended ses 0 -> N1 -> Ef -> M -> 0 (*) || | |f v v 0 -> N1 -> N2 -> N3 -> 0 called an _extension_ of M by N1. The construction continues: one sees that -> a lifting of f to N2 gives a splitting of the upper ses: Ef = M + N1. -> the upper sequence is split if and only if f lifts to N2. -> the set of extensions 0 -> N1 -> E -> M -> 0 (up to isomorphism of exact sequences) can be given the structure of an A-module Ext^1(N1,M), by multiplying the homomorphisms by elements of A and taking pullbacks. -> taking f in Hom_A(M,N3) to the extension Ef defines a continuation of the exact sequence 0 -> Hom_A(M,N1) -> Hom_A(M,N2) -> Hom_A(M,N3) -> Ext^1_A(M,N1) and this continues. Continuing this construction in explicit terms is possible, but would be long and tedious. Instead, we give a more systematic treatment in homological algebra. ==== Q. What are Ext^i good for? The functors Ext^i limit the depth -- e.g. as we have seen Hom(A/m, M) <> 0 implies depth = 0. As first consequence we get that lots of Ext^i = 0 is equivalent to depth > something. Q. Definition of Ext_A^r(M,N) via projective resolution of M More systematically, the functors Ext^i are right derived functor of Hom_A(M,?) Partly answers the old question: why do all that business about complexes and free resolutions? The connection with projective resolutions: if P is projective, a surjective map P -> M turns the composite with P -> M -> N3 into a morphism with a splitting P -> N2, together with worries about the kernel of where the kernel of P -> M goes. If we replace M with a projective resolution up to d terms, we get the derived functors Ext_A^i(M, ?) for i0. (3) Ext^i_A(M,N) is contravariant in M, and covariant in N. (4) s.e.s. of Ns gives rise to long exact cohomology seq. P. -> M a projective resolution (up to n terms, no need to assume finite for the moment) Define Ext^i_A(M,N) := H_i(Hom_A(P.,N)). because it Hom is contravariant in M, the indices increase Hom(Pi, N) -> Hom(P_{i+1}, N) -> .. so we write H_i. 1. Ext^0 = Hom 2. If M is already projective Ext^i(M,N) = 0 for all i. If M is projective, we can just take P. to be P0 = M and nothing more, so all Ext^i(M,N) = 0 for all i. More formally, we need to know that the definition is independent of the choice of P. Any two projective resolutions are homotopy equivalent. (Exercise, omit discussion.) 3. Nothing to prove 4. Let 0 -> N1 -> N2 -> N3 -> 0 be a s.e.s. Then each Pi is projective, so the sequence 0 -> Hom(Pi, N3) -> Hom(Pi, N2) -> Hom(Pi, N1) -> 0 is exact. Therefore the 3 complexes Hom(P.,Ni) fit together as a s.e.s. of complexes, and the l.e.s. comes by the usual snake lemma argument. ==== There is a very similar treatment of Ext^r(M,N) using _injective_ resolutions of N. The definition is "dual" to projective (in the categorical sense "just reverse the arrows"). Definition: An A-module I is injective if the contravariant functor Hom_A(?,I) is exact. A restatement is that whenever M1 into M2 is injective, every A-homomorphism f: M1 -> I is the restriction of some M2 -> I, or f _extends to_ M2. An Appendix below discusses what injective modules are, and gives a proof that every module M over a ring is a submodule of an injective module. However, you should treat this as an existence proof. (It is not comparable to the process of finding a free resolution by writing generators of a module and relations between them.) For an A-module N, there exists an injective resolution: a complex I.: 0 -> I0 -> I1 -> .. -> In .. (not assumed to be finite) such that -> each In is an injective A-module, -> the complex is exact at each i > 0, and -> H^0(I.) = N. In other words, there is an inclusion N into injective I0 then an inclusion of (I0/N) into injective I1, and so on indefinitely Now Ext^r(M,N) is the homology H_r(M,I.). This construction has a list of properties similar to the above, including a long exact sequence for s.e.s. in the M variable (Hom(?,N) covariant). That the 2 constructions give the same groups is left as an exercise in [Ma], Appendix B, p. 277. ==== There is an interplay between depth M. number of initial Ext^i(A/I,M) = 0. number of final H_{n-i}(K.(y1,..yn,M)) = 0 in Koszul complex. The basic point to remember is that a finite module M over a Noetherian ring A has I-depth zero if and only every f in I is a zerodivisor, and this happens iff there is a prime P in V(I) with P in Ass M. This holds because every zerodivisor of M is contained in an associated prime of M, of which there are finitely many, and I contained in a finite union of primes implies I is contained in one of the primes. So I-depth M = 0 <=> Hom(A/P,M) <> 0 for some P in Ass M. If A,m is local then m-depth M = 0 <=> M contains a submodule isomorphic to the field A/m. [Ma] Theorems 16.6-16.7 characterise depth of a finite module in terms of Ext^i: The I-depth of M is defined as the maximal length of a M-regular sequence (x1,.. xd) contained in I. This is equal to the number of initial Ext^i(A/I, M) that vanish (for i = 0,.. d-1). A ring and I an ideal. M a finite A-module. For given d, equivalent conditions: [Ma] Theorem 16.6 A Noetherian ring, I its ideal and M a finite A-module. Given d > 0, equivalent conditions: (1) Ext^i(N,M) = 0 for i = 0,..d-1 for EVERY finite A-module N with Supp N contained in V(I). (2) Ext^i(A/I,M) = 0 for i = 0,..d-1. (2') Ext^i(N,M) = 0 for i = 0,..d-1 for SOME A-module N with Supp N = V(I) (3) M has I-depth M >= d (<=> exists an M-regular sequence (y1,..yd) in I). See below for reminder on Supp and Ass. Assume N has property (2'). The main claim is i=0, Hom(N,M) = 0 I-depth M > 0. If I-depth M = 0 then every element of I is a zerodivisor of M. Every zerodivisor of M is contained in an associated prime P in Ass M. This is a finite set { Pi }, so I contained in Union Pi, which implies I contained in some P. Now the N has N_P <> 0, and a bit of localisation that I omit ([Ma] p. 129-130) shows eventually that Hom_A(N,M) <> 0, contradicting (2'). Hence there is an M-regular element x1 in I. The result holds for M/x1M by induction, so (2') implies (3). For the easier converse (3) implies (1), see [Ma] p. 130. ==== Summarise results in [Ma] about depth versus Exts. [not finished] A,m is local and N,M are finite A-modules Suppose dim N = r and Ext^s(N,M) <> 0. Then depth M <= r+s. Step 1. The basic case is r = 0 and s = 0, when statement reduces to N is 0-dimensional and phi: N -> M is nonzero then phi(N) subset Ass M, so depth M = 0. Step 2. In the statement, we can assume N = A/P with P a prime. By primary decomposition N has a finite descending filtration with Nj/N_{j+1} = A/Pj so dim A/Pj <= r. [Ma], Theorem 6.4 or [UCA], Theorem 7.6. Now by the long exact sequence of Exts, if Ext^s(N,M) <> 0 one of the A/Pj has Ext^s(A/Pj, M) <> 0. Step 3. If N = A/P and dim N = r > 0, choose x in P notin m, and do 0 -> N -x-> N -> Nbar -> 0. Then dim Nbar <= r-1. The exact sequence of Exts (remember, contravariant and coboundary does i |-> i+1) etc. by induction. ==== Intro to CM and Gor Let A, M be local. One of the characterisations of dimension: a _system of parameters_ (s.o.p.) is a sequence x1,.. xn in m that generate an m-primary submodule. This means that M/(x1,.. xn)M is an Artinian module, so of finite length or zero dimensional. Then dim M = min length of a s.o.p. We define A is Cohen-Macaulay if it has m-depth A = n = dim A. Thus A has a s.o.p. of length n = dim A that is a regular sequence. In geometric terms, we can cut A down by a regular sequence to an Artinian quotient ring, with each step the quotient by a principal ideal. The context: S polynomial ring k[x1,..xN] with maximal ideal m = (x1,..xN), either graded or localised at m. We can think of S as functions on the ambient space Spec S = AA^N. Much the same applies to S a regular local ring of dimension N. The ring A = S/I is the main object of study. In geometric situations we can think of it as functions on a subvariety V(I) in AA^N. If A has dim n, we can think of cutting it down by a sequence of parameters A/(x1,.. xn) to get to the Artinian quotient A/(x1,.. xn), or V(I) intersect (x1=.. xn=0) is zero dimensional. In this context, Cohen-Macaulay means x1,..xn is a regular sequence. [Gorenstein means that plus a bit.] The module A/(x1,..xn) of course depends on the regular s.o.p. you choose -- for example, we should be able to do the exercise of proving that l( A/(x1^s,x2,..xn) ) = s*l( A/(x1,x2,..xn) ) has length s times l(A/x1,..xn). However, the point is that the condition that the s.o.p. is a regular sequence is independent of the choice. Nor does the length, or dimension over k = A/m of the final Hom_A(k,A/(x1,x2,..xn) = Ext_A^n(k,A). Cohen-Macaulay ring: dimension n, and there exists a s.o.p. y1,.. yn for A that is a regular sequence, so that also => depth A = n The main result is that if this holds, every s.o.p. is a regular sequence. Cohen-Macaulay module: either M = 0, or dim M = n there exists a s.o.p. y1,.. yn for M that is a regular sequence The Gorenstein condition is the additional requirement that the Artinian quotient N = A/(x1,..xn) has socle of length 1. Here the socle of a module M is the set of elements g in M with Ann g = m. This is the same thing as Hom_M(A/m, M), or the biggest possible k-vector subspace of M. Consider A = S/I or M some A-module. how I is generated, free resolution of A, compared to codimension c = N-n system of parameters of A is defined as (y1,.. yn) in m that generate m-primary ideal. (one of the equivalent definitions of n = dim A). Note the two different steps: from S to quotient ring A, then from A to the Artinian quotient A/(x1,..xn). At the start, m is generated by the regular sequence x1,.. xN "unmixed" ideals. Take an ideal I = (x1,.. xr), and suppose that its height is r. (That is, every minimal prime of I has height r, or equivalently, no P in Ass A/I has dim A/P < r -- compare the counterexamples in Lect18-22, p.6-8.) Cohen-Mac or exactness of complexes only depends on dimension or height. Aim: Macaulay's unmixedness theorem depth is well-defined, independent of the choice of regular sequence Duality: Socle of the Artinian quotient is independent of regular sequence Cohen-Macaulay comes with a duality theory based on Ext_A^n(A,?) Gorenstein Artinian quotient Abar = A/(y1,.. yn) Socle of Abar: Hom_A(A/m, Abar) = { a in Abar | ma = 0 } = biggest k-vector space in Abar. ============ Appendix on Supp M and Ass M Intended as reminder/clarification. See [UCA] Chap 7 or [Ma] 6.4 A module M gives subsets of Spec A Supp M = { P | M_P <> 0 } That is, some a in M survives multiplication by every s notin P. Recall V(I) = { P | P contains I } in Spec A (any Zariski closed set). 1. For a cyclic module, Supp(A/I) = V(I). 2. For a finite module Supp N = V(Ann N) = V(rad(Ann N)). 3. Supp(N) contained in V(I) iff I contained in rad(Ann(N)). Proof of 1. Obviously, I = Ann(1 in A/I). If P contains I then s notin P implies s notin I, so multiplication by s takes 1 in A/I to a nonzero element of A/I. Localisation makes S = A-P invertible, so 1 survives multiplication by every s notin P, that is, (A/I)_P = S^-1(A/I) <> 0. Conversely, if P does not contain I then some s in I notin P and multiplication by s kills A/I. Proof of 2. Write N = sum A*n_i with generators ni. Then Supp N = Union Supp A*ni = Union V(Ann n_i) = V( Intersection Ann ni) = V(Ann N). Proof of 3. If J = Ann(N) then Supp(N) = V(J). Now V(J) contained in V(I) is equivalent to I contained in rad(J). Ass M is defined as the set of primes P such that M contains a copy of the integral domain A/P. If M is finite and <> 0, this is a finite and nonempty set. Every element a in A that is a zerodivisor for M is contained in one of the P in Ass M. ============ Appendix on injective modules Some details on their properties and existence Summary: For ZZ-modules, the inverse p-torsion modules (ZZ[1/p])/ZZ are injective, and every ZZ-module M embeds into a product of these (usually infinite). View any ring R as a ZZ-algebra; then for an injective ZZ-module I, the ZZ-module Hom_ZZ(R, I) becomes an R-module under premultiplication, and is an injective R-module. For an R-module M, view M as a ZZ-module and embed it into an injective ZZ-module I. An inclusion of M into an injective R-module is then provided by the tautological identity Hom_ZZ(M, I) = Hom_R(M, Hom_ZZ(R, I)). This is mostly cribbed from Charles A. Weibel, An introduction to homological algebra, CUP 1994. ==== Definition An R-module I is injective if Hom_R(-, I) is an exact functor. This means that if 0 -> M1 -> M2 is exact (that is, M1 embeds in M2) then any homomorphism e: M1 -> I has some extension f: M2 -> I. Extension means that f gives the same value as e on M1 in M2. Example. A k-vector space V is an injective k-module. This means that for U in W vector spaces, a k-linear map U -> V extends to W. (This requires Zorn's lemma.) Proposition To guarantee that a ZZ-module I is injective, it is enough to prove that any homomorphism e from an ideal J in R to I extends to a homomorphism f: R -> I. (This needs Zorn's lemma.) Proof Let M in N be an inclusion of R-modules and e: M -> I. Suppose that it has been extended to e': M' -> I for some M' with M in M' in N (start from M = M'). If b in N \ M', the extension from ideals allows me to extend e' to e": (M' + bR) -> I. For this, define J = { r in R s.t. br in M'}. Then J is an ideal of R. The homomorphism J -> I given by r |-> br |-> e'(br) is defined on J, so extends to R as a homomorphism f: R -> I. Then e": (M' + bR) -> I is defined as e' on M' and b -> f(1). In more detail, e": (m+br) |-> e'(m) + f(r). [To check well-defined: If some different m1, r1 has m+br = m1+br1 then b(r-r1) = m-m1 in M', so r-r1 in J where the extension f started, so f(r-r1) = e'(m-m1).] QED Corollary An Abelian group (a ZZ-module) is injective iff it is divisible. The modules QQ and (ZZ[1/p])/ZZ are injective, and any injective is a direct product of these. I refer to (ZZ[1/p])/ZZ as the inverse p-torsion module, by analogy with Macaulay's inverse monomials. Check that QQ/ZZ is the direct product of (ZZ[1/p])/ZZ taken over all primes p. An analogous construction works for a PID. Lemma For a ZZ-module M and nonzero m in M, there exists a prime p and a homomorphism f: M -> (ZZ[1/p])/ZZ with f(m) <> 0. Proof The annihilator of m in M is an ideal (n) in ZZ so mZZ iso ZZ/n. Choose any prime p | n and a surjective map ZZ/n ->> ZZ/p (if n = 0 then any prime p works). Compose with an embedding ZZ/p in (ZZ[1/p])/ZZ and extend from mZZ to the whole of M using injectivity of the module. QED Corollary Consider the set of all homomorphisms from M to the injective modules (ZZ[1/p])/ZZ. The direct sum of all these homomorphisms in an embedding of M into an injective ZZ-module. Now for modules over a general ring R (you need a little care about left and right modules if R is noncommutative). R is a ZZ-algebra. If A is a ZZ-module, the ZZ-module Hom_ZZ(R, A) becomes an R-module under premultiplication. Namely, r acts on Hom_ZZ(R, A) by f |-> fr, where fr is the map s -> f(sr) for s in R. That, multiply before applying the map (IMPORTANT). One checks the following points: 1. The identity Hom_ZZ(M, A) = Hom_R(M, Hom_ZZ(R, A)). holds for an R-module M and ZZ-module A. 2. If I is an injective ZZ-module then Hom_ZZ(R, I) is an injective R-module. 3. It follows that for any R-module M, the product of all homomorphisms M -> Hom_ZZ(R, (ZZ[1/p])/ZZ) is an embedding of M into an injective R-module. Finally, let F be a sheaf of OX-modules over a ringed space X. For P in X, the stalk F_P is an OX_P-module. Embed each stalk F_P into an injective OX_P-module I_P. This defines an OX-homomorphism of F into the sheaf of discontinuous sections of DisjointUnion I_P, which is an injective sheaf.