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Low-codimensional associated primes of graded components of local cohomology modules. (English) Zbl 1086.13006

Summary: Let \(R=\bigoplus_{n\geq 0}R_n\) be a homogeneous noetherian ring and let \(M\) be a finitely generated graded \(R\)-module. Let \(H_{R_+}^i(M)\) denote the \(i\)-th local cohomology module of \(M\) with respect to the irrelevant ideal \(R_+:= \bigoplus_{n>0}R_n\) of \(R\). We show that if \(R_0\) is a domain, there is some \(s\in R_0\setminus\{0\}\) such that the \((R_0)_s\)-modules \(H_{R_+}^i(M)\), are torsion-free (or vanishing) for all \(i\). On use of this, we can deduce the following results on the asymptotic behaviour of the \(n\)-th graded component \(H_{R_+}^i (M)_n\) of \(H_{R_+}^i(M)\) for \(n\to-\infty\):
If \(R_0\) is a domain or essentially of finite type over a field, the set \[ \{{\mathfrak p}_0\in\text{Ass}_{R_0}(H^i_{R_+}(M)_n)|\text{\,height}({\mathfrak p}_0)\leq 1\} \] is asymptotically stable for \(n\to-\infty\). If \(R_0\) is semilocal and of dimension 2, the modules \(H^i_{R_+}(M)\) are tame. If \(R_0\) is in addition a domain or essentially of finite type over a field, the set \(\text{Ass}_{R_0}(H^i_{R_+}(M)_n)\) is asymptotically stable for \(n\to-\infty\).

MSC:

13D45 Local cohomology and commutative rings
13E05 Commutative Noetherian rings and modules
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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[1] Brodmann, M.; Fumasoli, S.; Tajarod, R., Local cohomology over homogeneous rings with one-dimensional local base ring, Proc. Amer. Math. Soc., 131, 2977-2985 (2003) · Zbl 1041.13012
[2] Brodmann, M.; Hellus, M., Cohomological patterns of coherent sheaves over projective schemes, J. Pure Appl. Algebra, 172, 165-182 (2002) · Zbl 1011.13009
[3] Brodmann, M.; Katzman, M.; Sharp, R. Y., Associated primes of graded components of local cohomology modules, Trans. Amer. Math. Soc., 354, 11, 4261-4283 (2002) · Zbl 1013.13009
[4] Brodmann, M.; Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications, (Cambridge Stud. Adv. Math., vol. 60 (1998), Cambridge Univ. Press) · Zbl 0903.13006
[5] Eisenbud, D., Commutative Algebra with a View Towards Algebraic Geometry (1996), Springer-Verlag: Springer-Verlag New York
[6] Katzman, M., An example of an infinite set of associated primes of a local cohomology module, J. Algebra, 252, 161-166 (2002) · Zbl 1083.13505
[7] Katzman, M.; Sharp, R. Y., Some properties of top graded local cohomology modules, J. Algebra, 259, 599-612 (2003) · Zbl 1044.13008
[8] C.S. Lim, Graded local cohomology modules and their associated primes: the Cohen-Macaulay case, preprint; C.S. Lim, Graded local cohomology modules and their associated primes: the Cohen-Macaulay case, preprint · Zbl 1025.13004
[9] C.S. Lim, Graded local cohomology modules and their associated primes, Comm. Algebra, in press; C.S. Lim, Graded local cohomology modules and their associated primes, Comm. Algebra, in press · Zbl 1100.13016
[10] C.S. Lim, Graded local cohomology and its associated primes, PhD Thesis, Michigan State University, 2002; C.S. Lim, Graded local cohomology and its associated primes, PhD Thesis, Michigan State University, 2002
[11] Marley, T., The associated primes of local cohomology modules of small dimension, Manuscripta Math., 104, 519-525 (2001) · Zbl 0987.13009
[12] T. Marley, J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, preprint; T. Marley, J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, preprint · Zbl 1042.13010
[13] Singh, A. K., \(p\)-torsion elements in local cohomology modules, (Math. Res. Lett., vol. 7 (2000)), 165-176 · Zbl 0965.13013
[14] A.K. Singh, I. Swanson, Associated primes of local cohomology modules and Frobenius powers, Internat. Math. Res. Notices, in press; A.K. Singh, I. Swanson, Associated primes of local cohomology modules and Frobenius powers, Internat. Math. Res. Notices, in press · Zbl 1094.13006
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