On The Faltings. Local-global Principle. Department of Mathematics, University of Tabriz and IPM. Home Page. Title Page. Contents.

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1 On The Faltings Local-global Principle Page 1 of 23 Department of Mathematics, University of Tabriz and IPM

2 Outline Faltings Local-global Principle for the finiteness of local cohomoloy modules Annihilation theorem Local-global Principle for indimension < s modules Page 2 of 23

3 An important theorem in local cohomology is Faltings local-global principle for the finiteness of local cohomology modules [4, Satz 1]. Theorem[4, Satz 1]. Let r be a positive integer and M a finitely generated R- module. Then the following statements are equivalent: 1. The R-module H i a(m) is finitely generated for all i r, 2. The R p -module H i ar p (M p ) is finitely generated for all i r and for all p Spec R. Page 3 of 23

4 Another formulation of Faltings local-global principle: f a (M) = inf{f arp (M p ) p SpecR and dim R/p 0} Page 4 of 23 where f a (M) := inf{i N 0 H i a(m) is not finitely generated}.

5 Definition. Let n be a non-negative integer. to a is defined by The nth finiteness dimension f n a (M) of M relative f n a (M) := inf{f arp (M p ) p Supp (M/aM) and dim R/p n}. Page 5 of 23 Note that f n a (M) is either a positive integer or and that f 0 a (M) = f a (M). This notion has been introduced by Bahmanpour, Naghipour and Sedghi in [2].

6 We have the following formula: f 0 a (M) = inf{i N : H i a(m) is not finitely generated} f 1 a (M) = inf{i N : H i a(m) is not minimax} And if R is a semilocal ring, then f 2 a (M) = inf{i N : H i a(m) is not weakly laskerian} Page 6 of 23 Question. For each n N 0, what is the f n a (M)?

7 Definition. Let s N 0. An R-module M is said to be in dimension < s generated submodule N of M such that dim SuppM/N < s. if there exists a finitely Examples. 1- If M = 0, then M is in dimension < 1; 2- Every finitely generated module is in dimension < 0; 3- Every minimax module is in dimension < 1; 4- Every weakly laskerian module is in dimension < 2; Page 7 of 23

8 Theorem. Let (R, m) be a complete local ring and n N 0. Then f n a (M) = inf{i N : H i a(m) is not in dimension < n}. Consequence. Let (R, m) be a complete local ring and n N 0. Then the set Ass( t i=0 Hi a(m)) n Page 8 of 23 is a finite set, where t = f n a (M).

9 Theorem. Let a be an ideal of R, M be a finitely generated R-module and s, n N 0. the following statement are equivalent: Then 1- There is a t N 0 such that a t H i a(m) is in dimension < n for all i < s, 2- There is a t N 0 such that (ar p ) t H i ar p (M p ) is in dimension < n for all i < s and all p SpecR. Page 9 of 23

10 Question. Let a and b be ideals of R such that b a, M be a finitely generated R-module and s, n N 0. Are the following statements are equivalent? 1- There is a t N 0 such that b t H i a(m) is in dimension < n for all i < s, 2- There is a t N 0 such that (br p ) t H i ar p (M p ) is in dimension < n for all i < s and all p Spec R. Page 10 of 23

11 A generalization of Annihilation theorem Definition. Let a and b be ideals of R such that b a. The b-finiteness dimension of M relative to a is defined by f b a (M) = inf{i N 0 b Rad(0 : H i a(m))}. It is well-known that f b a (M) = inf{i N 0 b t H i a(m) 0 for all t N 0 } Page 11 of 23 = inf{i N 0 dim Supp b t H i a(m) 0 for all t N 0 }.

12 Definition. Let a and b be ideals of R such that b a. We define the nth b-finiteness dimension of M relative to a by f b a (M) n := inf{i N 0 dim Supp b t H i a(m) n for all t N 0 }. Page 12 of 23

13 Definition. Let a and b be ideals of R such that b a. The b-minimum a-adjusted depth of M is defined by λ b a(m) = inf{depth M p + ht (a + p)/p p Spec R \ V (b)}. It is easy to see that λ b a(m) = inf{λ br p ar p (M p ) p Spec R} = inf{λ br p ar p (M p ) p SpecR and dim R/p 0}. Page 13 of 23

14 Definition. depth of M by For a non-negative integer n, we define the nth b-minimum a-adjusted λ b a(m) n := inf{λ brp ar p (M p ) p Spec R and dim R/p n}. Page 14 of 23

15 Faltings Annihilation Theorem. If R is a homomorphic image of a regular ring, then, for every choice of the finitely generated R-module M and for every choice of the ideals a, b of R with b a, we have f b a (M) = λ b a(m). Theorem.[Khashyarmanesh and Salarian, 2004] If R is a homomorphic image of a Gorenstein ring, then, for every choice of the finitely generated R-module M and for every choice of the ideals a, b of R with b a, we have fa b (M) = λ b a(m). Page 15 of 23

16 Theorem. Assume that R is a Gorenstein ring. Let a and b be ideals of R such that b a, and let M be a finitely generated R-module. Then, for every non-negative integer n, fa b (M) n = λ b a(m) n. Page 16 of 23

17 Lemma 1. Let f : R R be a surjective homomorphism of Noetherian rings, and a and b be ideals of R such that b a. Let M be a finitely generated R -module. Then f b a (M ) n = f br ar (M ) n. Lemma 2. Let a and b be ideals of R such that b a. Let M be a finitely generated R-module, and let c be an ideal of R such that c (0 : M). Then Page 17 of 23 λ b a(m) n = λ (b+c)/c (a+c)/c (M) n.

18 Theorem. Assume that R is a homomorphic image of a Gorenstein ring. Let a and b be ideals of R such that b a, and let M be a finitely generated R-module. Then, for every non-negative integer n, fa b (M) n = λ b a(m) n. Page 18 of 23

19 Theorem. Assume that R is a homomorphic image of a Gorenstein ring. Let a and b be ideals of R such that b a, M be a finitely generated R-module and s, n N 0. statements are equivalent: Then the following 1- There is a t N 0 such that b t H i a(m) is in dimension < n for all i < s 2- f brp ar p (M p ) n for each p Spec R with dim R/p s, 3- For each p Spec R, there is a t N 0 such that (br p ) t H i ar p (M p ) is in dimension < n for all i < s. Page 19 of 23

20 Proof. 1 3) For each i with 0 i < n, let N i be a finitely generated submodule of b t H i a(m) such that dim Supp b t H i a(m)/n i < s. Let p Spec R. Since dim Supp (br p ) t H i ar p (M p )/(N i ) p dim Supp b t H i a(m)/n i < s, so, (br p ) t H i ar p (M p ) is in dimension < s. 3 2) Let p Spec R with dim R/p s. There is a maximal ideal m of R such that p m and dim R m /pr m s. By (3), there exists t N 0 and a finitely generated submodule N i of b t H i a(m) such that dim Supp (br m ) t H i ar m (M m )/(N i ) m = dim Supp (b t H i a(m)/n i ) m < s, for all i < n. So, (b t H i a(m)) p ((b t H i a(m)) m ) prm is finitely generated as R p -module. Since (b t H i a(m)) p is ar p -torsion, so there exists t N such that (b t+t H i a(m)) p (b t a t H i a(m)) p = 0. Therefore, f br p ar p (M p ) n. Page 20 of 23

21 2 1) Now, let R is Gorenstein. Then, f b a (M) s = inf{λ brp ar p (M p ) p Spec R and dimr/p s} = inf{f brp ar p (M p ) p Spec R and dimr/p s} n. So, There exists t N 0 such that for each i < n, dim Supp b t H i a(m) < s. Hence, b t H i a(m) is in dimension < s for all i < n. Page 21 of 23

22 References [1] D. Asadollahi and R. Naghipour, Faltings local-global principle for the niteness of local cohomology modules, Comm. Algebra, to appear. [2] K. Bahmanpour, R. Naghipour and M. Sedghi, Minimaxness and cofiniteness properties of local cohomology modules, Comm. Algebra, 41(2013), [3] M. R. Doustimehr and R. Naghipour, On the generalization of Faltings Annihilation Theorem, Arch. Math, to appear. Page 22 of 23 [4] G. Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann. 255(1981),

23 Thanks for your attention Page 23 of 23