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We start with the abstract theory of triangulated categories and triangulated functors. Because the octahedral axiom plays no role in our book, we give it minimal attention.

Then we introduce the homotopy category K(A,M), which has the same objects as C(A,M), and its morphisms are the degree 0 cohomology classes of the morphisms of C(A,M). We prove that the homotopy category K(A,M) is triangulated: its distinguished triangles are the images of the standard triangles in Cstr(A,M). A DG functor F as above induces a triangulated functor F : K(A,M) → K(B,N).Finally, we put a triangulated structure on the opposite homotopy category K(A,M)op, and we discuss contravariant triangulated functors.

There have been remarkably few systematic expositions of the theory of derived categories since its inception in the work of Grothendieck and Verdier in the 1960s. This book is the first in-depth treatment of this important component of homological algebra. It carefully explains the foundations in detail before moving on to key applications in commutative and noncommutative algebra, many otherwise unavailable outside of research articles. These include commutative and noncommutative dualizing complexes, perfect DG modules, and tilting DG bimodules. Written with graduate students in mind, the emphasis here is on explicit constructions (with many examples and exercises) as opposed to axiomatics, with the goal of demystifying this difficult subject. Beyond serving as a thorough introduction for students, it will serve as an important reference for researchers in algebra, geometry and mathematical physics.

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5 - Triangulated Categories and Functors

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5 - Triangulated Categories and Functors

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Derived Categories