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Local homology and cohomology on schemes. (English) Zbl 0894.14002
Let \(X\) be a quasi-compact separated scheme, and let \(X = \bigcup X_\alpha\) be its open covering. Suppose that a closed subscheme \(Z\) is proregular embedded in \(X.\) Thus, the defining ideal of \(Z\) is generated by a proregular sequence of sections from \(\bigcup \Gamma(X_\alpha, {\mathcal O}_{X_\alpha})\) [see J. P. C. Greenless and J. P. May, J. Algebra 149, No. 2, 438-453 (1992; Zbl 0774.18007)]. In the case where \(X\) is noetherian any closed subscheme is proregular embedded in \(X.\) For the pair \((X,Z)\) the authors describe a universal functorial duality expressed in terms of the right and left derived of the homomorphism and completion functors, respectively, which gives a sort of adjointness between the local cohomology and local homology supported in \(Z.\) In fact, using these results the authors generalize GM-duality (loc.cit.), the Peskine-Szpiro duality sequence [C. Peskine and L. Szpiro, Publ. Math., Inst. Hautes Étud. Sci. 42 (1972), 47-119 (1973; Zbl 0268.13008)], affine and formal duality theorems of Hartshorne [R. Hartshorne, “Residues and duality”, Lect. Notes Math. 20 (1996; Zbl 0212.26101)], and others.

MSC:
14B15 Local cohomology and algebraic geometry
32C37 Duality theorems for analytic spaces
14B20 Formal neighborhoods in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
18E30 Derived categories, triangulated categories (MSC2010)
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