## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 51, Number 1 (2007), 209-236.

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

Joseph Lipman and Amnon Neeman

#### Abstract

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.

#### Article information

**Source**

Illinois J. Math., Volume 51, Number 1 (2007), 209-236.

**Dates**

First available in Project Euclid: 20 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258735333

**Digital Object Identifier**

doi:10.1215/ijm/1258735333

**Mathematical Reviews number (MathSciNet)**

MR2346195

**Zentralblatt MATH identifier**

1124.14003

**Subjects**

Primary: 14A15: Schemes and morphisms

#### Citation

Lipman, Joseph; Neeman, Amnon. Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor. Illinois J. Math. 51 (2007), no. 1, 209--236. doi:10.1215/ijm/1258735333. https://projecteuclid.org/euclid.ijm/1258735333