A NOTE ON THE COFINITENESS OF LOCAL COHOMOLOGY MODULES FOR A PAIR OF IDEALS

Abstract

The cofinite property plays an important role in commutative algebra. R. Hartshorne (1970) posted the question: For which rings  and ideals  are the modules  is cofinite for all finitely generated modules ? A similar question is raised for local cohomology modules  w.r.t a pair of ideals . The first aim of the note is to build a new class of modules  and investigate its important properties in relation to the cofinite property of the module  by establishing a Grothendieck spectral sequence for the module . The second aim of the note is to prove the cofinite property of the module  under some conditions by applying the module class . With the assumption that  is an module such that  is finitely generated for all , this note has the first main result  if and only if , the second main result asserts that  is cofinite when  is a principal ideal and  is the module in dimension . These results are more extensive than some previous results because the first result is for the extended class of local cohomology modules, and the second results for the module  which is not necessary finitely generated.