Derived Categories in Representation Theory

VVe survey recent methods of derived categories in the representaion theory of al-gebras. 1Triangulated Categories and Brown Representabil-ity Definition 1.1 A triangulated category $\mathrm{C} $ is an additive $catego?\backslash y $ together with $(’l) $ an autofunctor $T:\mathrm{C} $ $arrow \mathrm{C}\sim $ ($i.e $. there is $T^{-1} $ such that $T\mathrm{o}T^{-1}=T^{-1}\mathrm{o}T=1_{C} $) called the translation, and $(_{\sim}^{u},’) $ a collection $\mathcal{T} $ of sextuples $(X, Y, Z, u, v, w) $: $Xuarrow Yarrow Zv $ $arrow T(X)\omega$ called (distinguished) triangles. These data are subject to the following four axioms: (TR1) (1) Every sextuple $(X, Y, Z,u, v, w) $ which is isomorphic to $a $ (distinguished) $tr\cdot-$ angle is $a $ (distinguished) triangle. (2) Every morphism $u:Xarrow Y $ is embedded in $a $ (distinguished) triangl