Affiliation:
1. 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 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.
Reference26 articles.
1. Local homology and cohomology on schemes;Lipman;Ann. Sci. École Norm. Sup.,1997
2. Brodmann, M.P., and Sharp, R.Y. (1998). Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press. Cambridge Studies in Advanced Mathematics.
3. Grothendieck, A. (1967). Local Cohomology, Springer. Lecture Notes in Math.
4. Problems on local cohomology;Eisenbud;Free Resolutions in Commutative Algebra and Algebraic Geometry, Sundance,1992
5. On the vanishing of local cohomology modules;Huneke;Inv. Math.,1990