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Local Cohomology and the Cohen-Macaulay Property

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Local Cohomology and the Cohen-Macaulay Property Local Cohomology and the Cohen-Macaulay property Abstract These lectures represent an extended version of the contents of a one hour introductory talk prepared by Florian Enescu and Sara Faridi for the Minnowbrook workshop to assist the lectures of one of the main speakers, Paul Roberts. Part of this talk was given earlier at the 2004 Utah mini-course on Classical problems in commutative algebra by the first author. The notes for that talk have been been typed and prepared by Bahman Engheta and we used part of them quite extensively in preparing this version. We would like to thank Bahman for his work. The references listed at the end were used in depth in preparing these notes and the authors make no claim of originality. Moreover, the reader is encouraged to consult these references for more details and many more results that were omitted due to time constraints. 1 Injective modules and essential extensions Throughout, let R be a commutative Noetherian ring. Definition 1 (injective module). An R-module E is called injective if one of the following equivalent con- ditions hold. 1. Given any R-module monomorphism f : N → M, every homomorphism u : N → E can be extended to a homomorphism g : M → E, i.e. g f = u. f 0 w NMw u g u E 2. For each ideal J of R, any homomorphism u : J → E can be extended to a homomorphism R → E. 3. The functor Hom( ,E) is exact. Example 2. From part 2 of Definition 1 it follows that Q and Q/Z are injective Z-modules. Definition 3 (divisible module). An R-module M is called divisible if for every m ∈ M and M-regular element r ∈ R, there is an m0 ∈ M such that m = rm0. Exercise 4. An injective R-module is divisible. The converse holds if R is a principal ideal domain; see the example above. A note on existence: If R → S is a ring homomorphism and E is an injective R-module, then HomR(S, E) is an injective S-module. [The S-module structure is given by s·ϕ( ) := ϕ(s ).] In particular, HomZ(R, Q) is an injective R-module and any R-module can be embedded in an injective R-module. We now show how to embed a given module into a minimal injective module. Definition 5 (essential extension). Let M ⊆ N be R-modules. We say that N is an essential extension of M if one of the following equivalent conditions holds: a) M ∩ N 0 6= 0 ∀ 0 6= N 0 ⊆ N. 1 b) ∀ 0 6= n ∈ N ∃ r ∈ R such that 0 6= rn ∈ M. ϕ c) ∀ N → Q, if ϕ|M is injective, then ϕ is injective. If additionally M 6= N, then we say that N is a proper essential extension of M. More generally, an injective map M →h N is called an essential extension if in the conditions above we replace M by h(M). Example 6. If R is a domain and Q(R) its fraction field, then R ⊆ Q(R) is an essential extension. Example 7. Let (R, m, k) be a local ring and N an R-module such that every element of N is annihilated by some power of the maximal ideal m. Let Soc(N) := AnnN (m) denote the socle of N (which is a k-vector space). Then Soc(N) ⊆ N is an essential extension. Exercise 8. An R-module N is Artinian if and only if Soc(N) is finite dimensional (as a k-vector space) and Soc(N) ⊆ N is essential. Exercise 9. Let M be a submodule of N. Use Zorn’s lemma to show that there is a maximal submodule N 0 ⊆ N containing M such that M ⊂ N 0 is an essential extension. Definition 10 (maximal essential extension). If M ⊆ N is an essential extension such that N has no proper essential extension, then M ⊆ N is called a maximal essential extension. The following proposition characterizes injective modules in terms of essential extensions. Proposition 11. a) An R-module E is injective if and only if it has no proper essential extension. b) Let M be an R-module and E an injective R-module containing M. Then any maximal essential extension N of M with N ⊆ E is a maximal essential extension. In particular, N is injective and thus a direct summand of E. c) If M ⊆ E and M ⊆ E0 are two maximal essential extensions, then there is an isomorphism E =∼ E0 that fixes M. Definition 12 (injective hull). A maximal essential extension of an R-module M is called an injective hull of M, denoted ER(M). Proposition 11 states that every module M has a unique injective hull up to isomorphism. Moreover, if M ⊆ E where E is an injective module, then M has a maximal essential extension ER(M) that is contained in E, and ER(M) is a direct summand of E. −1 0 Definition 13 (injective resolution). Let M be an R-module. Set E := M and E := ER(M). Induc- n En−1 tively define E := ER( /im(En−2)). Then the acyclic complex 0 1 n I : 0 → E → E → · · · → E → · · · is called an injective resolution of M, where the maps are given by the composition n−1 n−1 n−2 n−1 n−2 E → E /im(E ) ,→ ER(E /im(E )). Conversely, an acyclic complex I of injective R-modules is a minimal injective resolution of M if • M = ker(E0 → E1), 0 • E = ER(M), n n−1 n • E = ER(im(E → E )). For a Noetherian ring R one can give a specific description of each Ei appearing in the injective resolution of M as a direct sum of ER(R/p) for p ∈ SpecR. We refer interested reader to the sources mentioned in the references for more on this. 2 2 Local cohomology S n Definition 14 (I-torsion). Given an ideal I ⊆ R and an R-module M, set ΓI (M) := n(0 :M I ). The covariant functor ΓI ( ) over the category of R-modules is called the I-torsion functor, and for a homomor- phism f : M → N, ΓI (f) is given by the restriction f . ΓI (M) Proposition 15. ΓI ( ) is a left exact functor. Proof. Let f g 0 → L → M → N → 0 be a short exact sequence. We want to show that ΓI (f) ΓI (g) 0 → ΓI (L) −→ ΓI (M) −→ ΓI (N) is an exact sequence. Exactness at ΓI (L): ΓI (f) is injective as it is the restriction of the injective map f. Exactness at ΓI (M): It is clear that im(ΓI (f)) ⊆ ker(ΓI (g)). Conversely, let m ∈ ker(ΓI (g)). Then m ∈ ker(g) and therefore k m = f(l) for some l ∈ L. It remains to show that l ∈ ΓI (L). As m ∈ ΓI (M), we have I m = 0 for some k k k k integer k. Then f(I l) = I f(l) = I m = 0. As f is injective, I l = 0 and l ∈ ΓI (L). √ √ Exercise 16. ΓI = ΓJ if and only if I = J. i Definition 17 (local cohomology). The i-th local cohomology functor HI ( ) is defined as the i-th right derived functor of ΓI ( ). More precisely, given an R-module M, let I be an injective resolution of M: 0 1 n 0 d 1 d n d I : 0 → E → E → · · · → E → · · · Apply ΓI ( ) to I and obtain the complex: 0 1 n ΓI (I) : 0 → ΓI (E ) → ΓI (E ) → · · · → ΓI (E ) → · · · 0 i i i−1 i Then set HI (M) := ΓI (M) and HI (M) := ker(ΓI (d ))/ im(ΓI (d )) for i > 0. Note that HI ( ) is a covariant functor. i Clearly, if E is an injective R-module, then HI (E) = 0 for i > 0. Proposition 18. 1) If 0 → L → M → N → 0 is a short exact sequence, then we have an induced long exact sequence 0 0 0 0 → HI (L) → HI (M) → HI (N) → 1 1 1 HI (L) → HI (M) → HI (N) → · · · 2) Given a commutative diagram with exact rows: 0 w LMNw w w 0 u u u 0 w L0 w M 0 w N 0 w 0 then we have the following commutative diagram with exact rows: w i w i w i+1 w ··· HI (M) HI (N) HI (L) ··· u u u w i 0 w i 0 w i+1 0 w ··· HI (M ) HI (N ) HI (L ) ··· √ √ i i 3) I = J if and only if HI ( ) = HJ ( ). −1 −1 4) Localization: Let S ⊆ R be a multiplicatively closed set. Then S ΓI (M) = ΓS−1I (S M) and the same holds for the higher local cohomology modules. 3 2.1 Alternate construction of local cohomology n ∼ An alternate way of constructing the local cohomology modules: Consider the module HomR(R/I ,M) = n n m (0 :M I ). Now, if n > m, then one has a natural map R/I → R/I , forming an inverse system. Applying HomR( ,M), we get a direct system of maps: n ∼ [ n −→limn HomR(R/I ,M) = (0 :M I ) = ΓI (M). n As one might guess (or hope), it is also the case that i n ∼ i −→limn ExtR(R/I ,M) = HI (M). This follows from the theory of negative strongly connected functors – see [R]. Definition 19 (strongly connected functors). Let R, R0 be commutative rings. A sequence of covariant {T i} R R0 functors i>0 from the category of -modules to the category of -modules is said to be negative (strongly) connected if (i) Any short exact sequence 0 → L → M → N → 0 induces a long exact sequence 0 → T 0(L) → T 0(M) → T 0(N) → T 1(L) → T 1(M) → T 1(N) → · · · (ii) For any commutative diagram with exact rows 0 w LMNw w w 0 u u u 0 w L0 w M 0 w N 0 w 0 there is a chain map between the long exact sequences given in (i).
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