Filter regular sequences and endomorphisms of local cohomology modules

Abstract

Let R be a commutative Noetherian ring and a be an ideal of R such that cd(a,R) −<br><br>grade(a,R) ≤ 1, where cd(a,R) is the cohomological dimension of R with respect to<br><br>a and grade(a,R) is the grade of a. We show that whenever, Exti<br><br>R(Rz,R) = 0 =Ext_jR(Rz,Hn−1<br><br>a (R)) for all z ∈ a and all integers i, j with i &gt; 0 and j ≥ 0, then<br><br>there exists an exact sequence<br><br>0 → Extn<br><br>R(Hna (R), R) → EndR(Hna (R)) → EndR(Hn−1a (R)) → Extn+1R (Hn<br><br>a (R), R)of endomorphisms of local cohomology modules, where n := cd(a,R) and, for z ∈ R, Rz<br><br>is the ring of fractions of R with respect to multiplicatively closed subset {zj | j ≥ 0}<br><br>of R.