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Local Comology, Local Duality and Basic Notions of Tight Closure Theory

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Local Comology, Local Duality and Basic Notions of Tight Closure Theory LOCAL COMOLOGY, LOCAL DUALITY AND BASIC NOTIONS OF TIGHT CLOSURE THEORY Abstract. This lectures were given by Florian Enescu at the mini-course on classical problems in commutative algebra held at University of Utah, June 2004. The references listed were used extensively in preparing these notes and the author makes no claim of originality. Moreover, he encourages the reader to consult these references for more details and many more results that had to be omitted due to time constraints. This version has been typed and prepared by Bahman Engheta. 1. Injective modules, essential extensions, and local cohomology Throughout, let R be a commutative Noetherian ring. Definition. An R-module I is called injective if given any R-module monomor- phism f : N M, every homomorphism u : N I can be extended to a homo- morphism g :→M I, i.e. gf = u. → → f w 0 w N M u g u I Or equivalently, if the functor Hom( , I) is exact. Q Z Z Example. Q and / are injective -modules. Definition. An R-module M is called divisible if for every m M and M-regular element r R, there is an m′ M such that m = rm′. ∈ ∈ ∈ Exercise. An injective R-module is divisible. The converse holds if R is a principal ideal domain. A note on existence: If R S is a ring homomorphism and I is an injective → R-module, then HomR(S, I) is an injective S-module. [The S-module structure is Q given by s ϕ( ) := ϕ(s ).] In particular, HomZ(R, ) is an injective R-module and any R-module· can be embedded in an injective R-module. h Definition. An injective map M N is called an essential extension if one of the following equivalent conditions holds:→ a) h(M) N ′ =0 0 = N ′ N. b) 0 = ∩n N6 r∀ R6 such⊆ that 0 = rn h(M). ∀ 6 ϕ ∈ ∃ ∈ 6 ∈ c) N Q, if ϕ h is injective, then ϕ is injective. ∀ → ◦ 1 2LOCAL COMOLOGY, LOCAL DUALITY AND BASIC NOTIONS OF TIGHT CLOSURE THEORY Example. 1) If R is a domain and Q(R) its fraction field, then R Q(R) is an essential extension. ⊆ 2) Let M be a submodule of N. By Zorn’s lemma there is a maximal sub- module N ′ N containing M such that M N ′ is an essential extension. ⊆ ⊂ 3) Let (R, m, k) be a local ring and N an R-module such that every ele- ment of N is annihilated by some power of the maximal ideal m. Let Soc(N) := AnnN (m) denote the socle of N. Then Soc(N) N is an essential extension. (See exercise below.) ⊆ Note: The socle of a module N over a local ring (R, m, k)is a k-vector space. Exercise. 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. ⊆ Definition. If M N is an essential extension such that N has no proper essential extension, then M⊆ N is called a maximal essential extension. ⊆ Proposition. a) An R-module I is injective if and only if it has no proper essential extension. b) Let M be an R-module and I and injective R-module containing M. Then any maximal essential extension of M contained in I is a maximal essential extension. In particular, it is injective and thus a direct summand of I. c) If M I and M I′ are two maximal essential extensions, then there is ⊆ ⊆ ′ an isomorphism I ∼= I that fixes M. Definition. A maximal essential extension of an R-module M is called an injective hull of M, denoted ER(M). −1 0 Definition. Let M be an R-module. Set I := M and I := ER(M). Inductively − n In 1 define I := ER( /im(In−2)). Then the acyclic complex 0 1 n I : 0 I I I → → →···→ →··· is called an injective resolution of M, where the maps are given by the composition − − n−1 In 1 In 1 .( I / n−2 ֒ ER( / n−2 → im(I ) → im(I ) Conversely, an acyclic complex I of injective R-modules is a minimal injective resolution of M if M = ker(I0 I1), • 0 → I = ER(M), • n n−1 n I = ER(im(I I )). • → n I-torsion: Given an ideal I R and an R-module M, set ΓI (M) := (0 :M I ). ⊆ Sn Then ΓI ( ) defines a covariant functor and for a homomorphism f : M N,ΓI (f) is given by the restriction f . → I Γ (M) Proposition. ΓI ( ) is a left exact functor. LOCAL COMOLOGY, LOCAL DUALITY AND BASIC NOTIONS OF TIGHT CLOSURE THEORY3 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 m = f(l) for some l L. It ∈ ∈ k ∈ remains to show that l ΓI (L). As m ΓI (M), we have I m = 0 for some integer k k ∈ k ∈ k k. Then f(I l)= I f(l)= I m =0. As f is injective, I l = 0 and l ΓI (L). ∈ Exercise. ΓI =ΓJ if and only if √I = √J. i Definition. The i-th local cohomology functor HI ( ) is defined as the 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 I I I → → →···→ →··· Apply ΓI ( ) to I and obtain the complex: 0 1 n ΓI (I): 0 ΓI (I ) ΓI (I ) ΓI (I ) → → →···→ →··· 0 i i i−1 Then set HI ( ):=ΓI ( ) and HI (M) := ker(ΓI (d ))/ im(ΓI (d )) for i> 0. Note i that HI ( ) is a covariant functor. Proposition. 1) If 0 L M N 0 is a short exact sequence, then we have an induced long→ exact→ sequence→ → 0 H0(L) H0(M) H0(N) → I → I → I → H1(L) H1(M) H1(N) I → I → I →··· 2) Given a commutative diagram with exact rows: w w w 0 w L M N 0 u u u ′ ′ ′ w w w 0 w L M N 0 then we have the following commutative diagram with exact rows: i i i+1 w w w w H (M) H (N) H (L) · · · I I I · · · u u u i ′ i ′ i+1 ′ w w w w H (M ) H (N ) H (L ) · · · I I I · · · i i 3) √I = √J if and only if HI ( )= HJ ( ). 4LOCAL COMOLOGY, LOCAL DUALITY AND BASIC NOTIONS OF TIGHT CLOSURE THEORY A note on localization: Let S R be a multiplicatively closed set. Then −1 −1 ⊆ S ΓI (M)=ΓS−1I (S M) and the same holds for the higher local cohomology modules. i Clearly, if E is an injective R-module, then HI (E)=0 for i> 0. 2. Local cohomology An alternate way of constructing the local cohomology modules: Consider the n n module HomR(R/I ,M) = (0 :M I ). Now, if n m, then one has a natural map n m ∼ 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. Let R, R′ be commutative rings. A sequence of covariant functors i ′ T i> : R-modules R -modules is said to be negative (strongly) connected if { } 0 → (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 w w w 0 w L M N 0 u u u ′ ′ ′ w w w 0 w L M N 0 there is a chain map between the long exact sequences given in (i). 0 0 0 i i > Theorem. Let ψ : T U be a natural equivalence, where T i> , U i are → { } 0 { } 0 strongly connected. If T i(I)= U i(I)=0 for all i> 0 and injective modules I, then i i i there is a natural equivalence of functors ψ = ψ i> : T i U i. { } 0 { } →{ } i n i The above theorem implies that limn ExtR(R/I ,M) ∼= HI (M). −→ Remark. i) One can replace the sequence of In by any decreasing sequence n { } n of ideals Jt which are cofinal with I , i.e. t n such that I Jt { } n { } ∀ ∃ ⊆ and n t such that Jt I . ∀ ∃ i ⊆ i ii) Every element of HI (M) is killed by some power of I, as every m HI (M) i n n ∈ is the image of some ExtR(R/I ,M) which is killed by I . LOCAL COMOLOGY, LOCAL DUALITY AND BASIC NOTIONS OF TIGHT CLOSURE THEORY5 x iii) For any x R, the homomorphism M · M induces a homomorphism ·x ∈ −→ Hi (M) Hi (M). I −→ I Proposition. Let M be a finitely generated R-module and I R an ideal.
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