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 \(n\to-\infty\).


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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