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

Author

Listed:
  • Pinger Zhang

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

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.

Suggested Citation

  • Pinger Zhang, 2024. "Local (Co)homology and Čech (Co)complexes with Respect to a Pair of Ideals," Mathematics, MDPI, vol. 12(3), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:437-:d:1329147
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