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All notation and conventions are from our notes on Derived Categories (DTC) or Derived Categories of Sheaves (DCOS). In particular we assume that every abelian category comes with canonical structures that allow us to define the cohomology of chain complexes in an unambiguous way. If we write complex we mean cochain complex. As usual we write A= 0 to indicate that A is a zero object (not necessarily the canonical one). We say that a complex is exact if all its cohomology objects are zero, and reserve the label acyclic for complexes described in (DTC2, Definition 4). Given a scheme X we have the abelian category Qco (X) of quasi-coherent sheaves on X, whose derived category we denote by Dqcoh (X). If X is concentrated (CON, Definition 3) then Qco (X) is grothendieck abelian (MOS, Proposition 66) so this class of schemes is prevalent when dealing with derived functors defined on quasi-coherent sheaves. This is a very mild condition to put on a scheme: for example, any noetherian or affine scheme is concentrated. In this note we develop the basic theory of the triangulated categories Dqcoh (X). The major theorems are as follows: