Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor

By J. Lipman, A. Neeman. 2006

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For a map f : X → Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint f of Rf∗ respects small direct sums. This is equivalent to the existence of a functorial isomorphism f×OY ⊗ L Lf(− ) −→ ∼ f(−); to quasi-properness (preservation by Rf∗ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of Lf (finite tor-dimensionality), or of the functor f; and to some other conditions. We use a globalization, previously known only for divisorial schemes, of the local definition of pseudo-coherence of complexes, as well as a refinement of the known fact that the derived category of complexes with quasi-coherent homology is generated by a single perfect complex.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e2e130b9076f03258ea4c4246758840c24e87436

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Citations in the corpus

Orbifold techniques in degeneration formulas (2011, background): “−→ p∗L K/T. The morphism pis proper. It is also l.c.i., therefore Gorenstein, and its dualizing complex ωp is a line bundle positioned in degree −1. By ORBIFOLD TECHNIQUES IN DEGENERATION FORMULAS 73 [LN] the functor Rp∗ : D(C) → D(K) has a right adjoint p! which is isomorphic to the functor F→ Lp∗(”
Strong generators in $D^{perf}(X)$ and $D^b_{coh}(X)$ (2017, background): “2], but the nonnoetherian generality of the statement and proof in [21] might be confusing.”
SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES (2014, background): “ Ñ X and take the complex Rg˚Or C0 . Since g is proper being a composition of proper morphisms, the complex Rg˚Or C0 is a cohomologically bounded complex with coherent cohomology. From Theorem 4.1 of [LN] we know that, given any integer m, we may ﬁnd a perfect complex P¨ and a morphism u : P¨ Ñ Rg˚O”
Derived equivalence for quantum symplectic resolutions (2011, unknown): “1 itself is partly inspired by [44].”