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J. Noncommut. Geom., 2013, Volume 7, Issue 2, Pages 357–371 (Mi jncg1)  

Does full imply faithful?

A. Canonacoa, D. Orlovb, P. Stellaric

a Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy
b Steklov Mathematical Institute, ul. Gubkina 8, 119991, Moscow, Russian Federation
c Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milano, Italy

Abstract: We study full exact functors between triangulated categories. With some hypotheses on the source category we prove that it admits an orthogonal decomposition into two pieces such that the functor restricted to one of them is zero while the restriction to the other is faithful. In particular, if the source category is either the category of perfect complexes or the bounded derived category of coherent sheaves on a noetherian scheme supported on a closed connected subscheme, then any non-trivial exact full functor is faithful as well. Finally we show that removing the noetherian hypothesis this result is not true.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00336
11-01-00568
Ministry of Education and Science of the Russian Federation 5139.2012.1
11.G34.31.0023
Italian Ministry of Education, University and Research PRIN 2008
The second author was partially supported by RFBR grants 11-01-00336, 11-01-00568, NSh grant 5139.2012.1, by AG Laboratory HSE, RF gov. grant, ag. 11.G34.31.0023. The third author was partially supported by the MIUR of the Italian Government in the framework of the National Research Project "Geometria algebrica e aritmetica, teorie coomologiche e teoria dei motivi" (PRIN 2008).


DOI: https://doi.org/10.4171/JNCG/120


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