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Spring 2007 Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor
Joseph Lipman, Amnon Neeman
Illinois J. Math. 51(1): 209-236 (Spring 2007). DOI: 10.1215/ijm/1258735333


For a map $f\colon X\to Y$ of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint $f^\times$ of $\mathbf Rf_*$ respects small direct sums. This is equivalent to the existence of a functorial isomorphism $f^\times\mathcal O_{Y}\otimes^{\mathbf L} \mathbf Lf^*(\<-\<)\! {\longrightarrow{}^\sim} f^\times (-)$; to quasi-properness (preservation by $\Rf$ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of $\mathbf Lf^*\<$ (finite tor-dimensionality), or of the functor $f^\times\<$; 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.


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Joseph Lipman. Amnon Neeman. "Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor." Illinois J. Math. 51 (1) 209 - 236, Spring 2007.


Published: Spring 2007
First available in Project Euclid: 20 November 2009

zbMATH: 1124.14003
MathSciNet: MR2346195
Digital Object Identifier: 10.1215/ijm/1258735333

Primary: 14A15

Rights: Copyright © 2007 University of Illinois at Urbana-Champaign

Vol.51 • No. 1 • Spring 2007
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