J. Noncommut. Geom., 2013, Volume 7, Issue 2, Pages 357–371
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
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.
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