Does full imply faithful?

TL;DR

This paper investigates conditions under which full exact functors between triangulated categories are faithful, showing that under certain hypotheses they decompose into faithful and zero parts, with counterexamples when hypotheses are removed.

Contribution

It proves that full exact functors often decompose into faithful and zero parts under specific conditions, extending understanding of their structure in triangulated categories.

Findings

Full functors decompose into faithful and zero parts under certain hypotheses

Non-trivial full functors are faithful for perfect complexes and bounded derived categories on noetherian schemes

Removing noetherian assumptions invalidates the faithfulness result

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.