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

Pinger Zhang ()
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Pinger Zhang: College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Mathematics, 2024, vol. 12, issue 3, 1-12

Abstract: 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.

Keywords: local cohomology; ?ech complex; Koszul complex; Matlis–Greenless–May equivalence; Grothendieck’s local duality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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