Local (Co)homology and Čech (Co)complexes with Respect to a Pair of Ideals.

Let I and J be two ideals of a commutative ring R. We introduce the concepts of the C ˇ ech complex and C ˇ ech cocomplex with respect to (I , J) and investigate their homological properties. In addition, we show that local cohomology and local homology with respect to (I , J) are expressed by the above complexes. Moreover, we provide a proof for the Matlis–Greenless–May equivalence with respect to (I , J) , which is an equivalence between the category of derived (I , J) -torsion complexes and the category of derived (I , J) -completion complexes. As an application, we use local cohomology and the C ˇ ech complex with respect to (I , J) to prove Grothendieck's local duality theorem for unbounded complexes. [ABSTRACT FROM AUTHOR]