Local Homology and Cohomology on Schemes

Details Local Homology and Cohomology on Schemes Local Homology and Cohomology on Schemes

1995-03-30

arxiv.org/abs/alg-geom/9503025v1

Mathematics

Commutative Algebra

DVI file pub/lipman/homology.dvi (214776K, 38 pages) available via anonymous ftp (binary) at ftp.math.purdue.edu, AMSLaTeX v 1

Scientific paper

We prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a "proregular" subscheme Z---for example, any separated noetherian scheme and any closed subscheme---there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, the i-th left-derived completion functor is the "local homology" sheaf $Ext^i(\R\Gamma_Z\O_X, -)$. Sheafified generalizations of a number of duality theorems scattered about the literature result, e.g., the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.

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