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Zbl 0894.14002
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Lipman, Joseph
Local homology and cohomology on schemes. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 30, No. 1, 1-39 (1997). ISSN 0012-9593

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 $\bigcup \Gamma(X_\alpha, {\cal O}_{X_\alpha})$ [see {\it J. P. C. Greenless} and {\it 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 [{\it C. Peskine} and {\it L. Szpiro}, Publ. Math., Inst. Hautes Étud. Sci. 42 (1972), 47-119 (1973; Zbl 0268.13008)], affine and formal duality theorems of Hartshorne [{\it R. Hartshorne}, ``Residues and duality'', Lect. Notes Math. 20 (1996; Zbl 0212.26101)], and others.
[A.G.Aleksandrov (Moskva)]

MSC 2000:: *14B15 Local cohomology
32C37 Duality theorems (analytic spaces)
14B20 Formal neighborhoods
14F20 Grothendieck cohomology and topology
18E30 Derived categories, etc.

Keywords: quasi-compact scheme; formal scheme; Koszul complex; proregular sequence; local cohomology; local homology; adjointness; derived functor; Bousfield localization; Matlis duality; Grothendieck duality; Warwick duality theorem; Peskine-Szpiro duality; functorial duality

Citations: Zbl 0212.26101; Zbl 0774.18007; Zbl 0268.13008

Cited in: Zbl 0953.14012

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