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

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Let I and J be two ideals of a commutative ring R. We introduce the concepts of the Cˇech complex and Cˇech 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 Cˇech complex with respect to (I,J) to prove Grothendieck’s local duality theorem for unbounded complexes.