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Greenlees-May duality of formal schemes. (English) Zbl 0953.14012
Alonso Tarrío, Leovigildo et al., Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes. Providence, RI: American Mathematical Society. Contemp. Math. 244, 93-112 (1999).
Let $$Z$$ be a closed subscheme of a noetherian scheme $$X$$. In a former paper [L. Alonso Tarrío, A. Jeremías López and J. Lipman, Ann. Sci. Ec. Norm. Supér., IV. Sér. 30, No. 1, 1-39 (1997; Zbl 0894.14002)], the authors inspired by a paper of J. P. C. Greenlees and J. P. May [J. Algebra 149, No. 2, 438-453 (1992; Zbl 0774.18007)] about duality between local cohomology and local homology for modules over a commutative ring, proved a result which basically states that on quasi-coherent complexes $$\mathcal F$$ the “homology localization” functor $$\text{RHom}^*({\mathbb R}\Gamma_Z{\mathcal O}_X,{\mathcal F})$$ is a left-derived functor of $$\Lambda_Z:=$$completion along $$Z$$, the corresponding map to $${\Lambda}_Z{\mathcal F}$$ being such that the composition with the natural map $${\mathcal F}\to\text{RHom}^*({\mathbb R}{\Gamma}_Z{\mathcal O}_X, \mathcal F)$$ is the completion $${\mathcal F}\to{\Lambda}_Z{\mathcal F}$$.
In the paper under review (which is the second of the series of the three papers of this volume) the authors extend the result to an arbitrary noetherian formal scheme $$\mathcal X$$, with “quasi-coherent” replaced by “direct limit of coherent”.
For the entire collection see [Zbl 0927.00024].

##### MSC:
 14F99 (Co)homology theory in algebraic geometry 18E25 Derived functors and satellites (MSC2010) 14B15 Local cohomology and algebraic geometry 32C37 Duality theorems for analytic spaces 13D45 Local cohomology and commutative rings 18G10 Resolutions; derived functors (category-theoretic aspects)