# zbMATH — the first resource for mathematics

Nonnoetherian Grothendieck duality. (English) Zbl 0953.14013
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, 115-123 (1999).
This paper is the third (and the last) of the series of papers of this volume. Let $$f:X\to Y$$ be a separated morphism of quasi-compact, quasi-separated schemes. The author sketches a proof of the fact that the functor $${\mathbb R}f_*:{\mathbb D}^+_{qc}(X)\to{\mathbb D}^+(Y)$$ has a right adjoint $$f^!$$. Moreover, if $$f$$ is proper, finitely presented and flat, then duality and tor-independent base-change hold for $$f^!$$. The novelty of this paper is that the noetherian assumption of the schemes in question is dropped.
For the entire collection see [Zbl 0927.00024].

##### MSC:
 14F99 (Co)homology theory in algebraic geometry 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 14A15 Schemes and morphisms